One Number Tells You How Many Answers an Equation Has Before You Solve It
Use the quadratic formula on any equation, compute the discriminant to decide whether the roots are real and distinct, equal or not real, find the value of k that forces equal roots, and solve speed, work and dimension problems that turn out quadratic.
What do you do when a quadratic simply will not factorise?
Try to solve by splitting the middle term. You need two numbers with product and sum , and no such pair of integers exists. The equation has perfectly good roots — they are just not rational, so no amount of searching will find them by inspection.
That is the gap this part of the chapter fills. The quadratic formula solves every quadratic equation, whether its roots are whole numbers, fractions, surds, or not real at all. Factorisation remains faster when it works; the formula never fails.
And buried inside the formula is something even more useful than the roots. The expression under the square root sign, , is called the discriminant, and its sign alone tells you the nature of the roots before you calculate them:
- Positive — two distinct real roots
- Zero — two equal real roots
- Negative — no real roots at all
So a question asking "does this equation have real roots?" needs one subtraction, not a full solution. **And a question asking for the value of that makes the roots equal becomes an equation in , formed by setting the discriminant to zero.
The part closes with the word problems the formula makes possible: boats in streams, taps filling tanks, trains changing speed, and right triangles with one side given in terms of another. All of them produce quadratics that rarely factorise neatly**, which is exactly why they are set here rather than in Part 1.
This page covers the second part of the CBSE Class 10 Maths chapter on quadratic equations: the quadratic formula, the discriminant and the nature of roots, conditions for equal roots, and applied problems.
That is the gap this part of the chapter fills. The quadratic formula solves every quadratic equation, whether its roots are whole numbers, fractions, surds, or not real at all. Factorisation remains faster when it works; the formula never fails.
And buried inside the formula is something even more useful than the roots. The expression under the square root sign, , is called the discriminant, and its sign alone tells you the nature of the roots before you calculate them:
- Positive — two distinct real roots
- Zero — two equal real roots
- Negative — no real roots at all
So a question asking "does this equation have real roots?" needs one subtraction, not a full solution. **And a question asking for the value of that makes the roots equal becomes an equation in , formed by setting the discriminant to zero.
The part closes with the word problems the formula makes possible: boats in streams, taps filling tanks, trains changing speed, and right triangles with one side given in terms of another. All of them produce quadratics that rarely factorise neatly**, which is exactly why they are set here rather than in Part 1.
This page covers the second part of the CBSE Class 10 Maths chapter on quadratic equations: the quadratic formula, the discriminant and the nature of roots, conditions for equal roots, and applied problems.
Formula
What is the quadratic formula and how do you apply it?
**Write the equation in standard form, identify , and with their signs, and substitute.**
for with . The two roots come from the two signs of the square root, which is why a quadratic has two answers.
Worked example 1 — rational roots. Solve .
Here , , , so
giving and .
**Check at **: . Correct.
Notice the discriminant was a perfect square, which is precisely why this equation could also have been factorised.
Worked example 2. Solve .
giving and .
**Check at **: . Correct.
Worked example 3 — irrational roots, where factorisation fails. Solve .
**The roots are and .
Two steps deserve attention here.** First, was simplified to — leaving it as costs a mark. Second, the common factor of was cancelled from all three terms of the fraction, not just from the two obvious ones. **Cancelling only the and the while forgetting the denominator is a standard slip.
Check the roots without a calculator.** Their sum should be , and indeed . Their product should be , and . Both relations hold, which is the cleanest possible verification of surd roots.
Worked example 4 — a surd coefficient. Solve .
A single repeated root, because the discriminant vanished.
Check: . Correct.
The three sign errors that account for most lost marks. Writing as when is already negative — for the numerator begins with . Writing without the minus in front. And forgetting that the whole numerator sits over , not just the square root.
for with . The two roots come from the two signs of the square root, which is why a quadratic has two answers.
Worked example 1 — rational roots. Solve .
Here , , , so
giving and .
**Check at **: . Correct.
Notice the discriminant was a perfect square, which is precisely why this equation could also have been factorised.
Worked example 2. Solve .
giving and .
**Check at **: . Correct.
Worked example 3 — irrational roots, where factorisation fails. Solve .
**The roots are and .
Two steps deserve attention here.** First, was simplified to — leaving it as costs a mark. Second, the common factor of was cancelled from all three terms of the fraction, not just from the two obvious ones. **Cancelling only the and the while forgetting the denominator is a standard slip.
Check the roots without a calculator.** Their sum should be , and indeed . Their product should be , and . Both relations hold, which is the cleanest possible verification of surd roots.
Worked example 4 — a surd coefficient. Solve .
A single repeated root, because the discriminant vanished.
Check: . Correct.
The three sign errors that account for most lost marks. Writing as when is already negative — for the numerator begins with . Writing without the minus in front. And forgetting that the whole numerator sits over , not just the square root.
How does the discriminant tell you the nature of the roots?
**Compute and look at its sign. You do not need the roots themselves.**
The discriminant is written or , and
- ** — the roots are real and distinct, and the square root of a positive number gives two different values
- — the roots are real and equal**, both being
- ** — the equation has no real roots, because no real number squares to a negative one
Worked example 1.** Find the nature of the roots of .
**Since , the equation has no real roots. Notice that no attempt to solve it was needed, and that trying to factorise it would have wasted several minutes.
Worked example 2.** Find the nature of the roots of .
The roots are real and equal, and both equal
Worked example 3. Find the nature of the roots of .
Real and distinct — and since is not a perfect square, the roots are irrational, which is the extra detail a careful answer adds.
That refinement is worth knowing. When :
- ** a perfect square and the coefficients rational — the roots are rational, so factorisation will work
- positive but not a perfect square — the roots are irrational**, and they occur as a conjugate pair such as
So the discriminant also predicts whether factorisation is worth attempting. A quick check of before starting tells you which method to use, and that is a genuine time-saver in an examination.
The graph behind the three cases. A quadratic's graph is a parabola, and:
- ** — the parabola cuts** the -axis at two points
- ** — it touches the axis at exactly one point
- — it misses the axis entirely
The connection is exact, not an analogy.** The roots are the -coordinates of the crossings, so the number of crossings is the number of real roots, and the discriminant is counting crossings.
One caution about the wording. "No real roots" is not the same as "no roots". The equation still has two roots — they are simply not real numbers, and Class 11 names them. At this level the required phrase is no real roots, and writing "no solution" is marked wrong.
The discriminant is written or , and
- ** — the roots are real and distinct, and the square root of a positive number gives two different values
- — the roots are real and equal**, both being
- ** — the equation has no real roots, because no real number squares to a negative one
Worked example 1.** Find the nature of the roots of .
**Since , the equation has no real roots. Notice that no attempt to solve it was needed, and that trying to factorise it would have wasted several minutes.
Worked example 2.** Find the nature of the roots of .
The roots are real and equal, and both equal
Worked example 3. Find the nature of the roots of .
Real and distinct — and since is not a perfect square, the roots are irrational, which is the extra detail a careful answer adds.
That refinement is worth knowing. When :
- ** a perfect square and the coefficients rational — the roots are rational, so factorisation will work
- positive but not a perfect square — the roots are irrational**, and they occur as a conjugate pair such as
So the discriminant also predicts whether factorisation is worth attempting. A quick check of before starting tells you which method to use, and that is a genuine time-saver in an examination.
The graph behind the three cases. A quadratic's graph is a parabola, and:
- ** — the parabola cuts** the -axis at two points
- ** — it touches the axis at exactly one point
- — it misses the axis entirely
The connection is exact, not an analogy.** The roots are the -coordinates of the crossings, so the number of crossings is the number of real roots, and the discriminant is counting crossings.
One caution about the wording. "No real roots" is not the same as "no roots". The equation still has two roots — they are simply not real numbers, and Class 11 names them. At this level the required phrase is no real roots, and writing "no solution" is marked wrong.
How do you find the value of k that gives equal roots?
**Set the discriminant equal to zero and solve the resulting equation in . Then reject any value that makes the equation stop being quadratic, and verify by substitution.
Worked example 1.** Find the values of for which has two equal roots.
Here , , , so
Setting :
Both values are admissible, since neither makes zero.
**Verify with .** The repeated root is
Substituting into the equation:
Correct, so the value of is right and the root really is repeated.
Worked example 2 — where one value must be rejected. Find the value of for which has two equal roots.
First put it in standard form:
so , , and
Setting :
Now the rejection. If the equation becomes , which is not a quadratic equation at all — indeed it is not an equation with any solution. **The definition requires , so is inadmissible and .
Verify with **: the equation becomes , or , which is . **A repeated root at , exactly as required.
That rejection is the point of the question.** Whenever the unknown letter sits in the coefficient of , check whether your answer makes that coefficient zero, and say in writing why such a value is discarded.
Worked example 3 — a condition for real roots rather than equal roots. Find the values of for which has real roots.
Real roots need , not :
The answer is an inequality, not a number, and that is the distinction most often missed in this family of questions.
The three conditions side by side, because questions switch between them without warning.
- "Equal roots" or "a repeated root" — set
- "Real roots" or "real and equal or distinct" — set
- "Two distinct real roots" — set
- "No real roots" — set
Read which one is being asked before writing anything. An equals sign where an inequality belongs turns a full answer into a single value.
Worked example 1.** Find the values of for which has two equal roots.
Here , , , so
Setting :
Both values are admissible, since neither makes zero.
**Verify with .** The repeated root is
Substituting into the equation:
Correct, so the value of is right and the root really is repeated.
Worked example 2 — where one value must be rejected. Find the value of for which has two equal roots.
First put it in standard form:
so , , and
Setting :
Now the rejection. If the equation becomes , which is not a quadratic equation at all — indeed it is not an equation with any solution. **The definition requires , so is inadmissible and .
Verify with **: the equation becomes , or , which is . **A repeated root at , exactly as required.
That rejection is the point of the question.** Whenever the unknown letter sits in the coefficient of , check whether your answer makes that coefficient zero, and say in writing why such a value is discarded.
Worked example 3 — a condition for real roots rather than equal roots. Find the values of for which has real roots.
Real roots need , not :
The answer is an inequality, not a number, and that is the distinction most often missed in this family of questions.
The three conditions side by side, because questions switch between them without warning.
- "Equal roots" or "a repeated root" — set
- "Real roots" or "real and equal or distinct" — set
- "Two distinct real roots" — set
- "No real roots" — set
Read which one is being asked before writing anything. An equals sign where an inequality belongs turns a full answer into a single value.
How do speed, tap and triangle problems become quadratic equations?
They become quadratic whenever the unknown appears in a denominator, or whenever two quantities multiply to give a third. Time is distance over speed, so a speed in a denominator produces a quadratic; areas and the Pythagoras theorem produce them directly.
Worked example 1 — a boat in a stream. The speed of a boat in still water is km/h. It goes km upstream and returns downstream to the starting point in hours minutes. Find the speed of the stream.
Let the stream's speed be km/h, so the upstream speed is and the downstream speed is . The total time is hours:
Combining the fractions:
so or . **A speed cannot be negative, so the stream flows at km/h.
Check**: upstream at km/h takes hours, downstream at km/h takes hours, and the total is hours. Correct.
Notice that the numerator simplified beautifully because the terms cancelled. That happens whenever the same distance is covered both ways, and spotting it saves a page of algebra.
Worked example 2 — two taps. Two water taps together can fill a tank in hours. The larger tap takes hours less than the smaller one to fill the tank alone. Find the time each takes separately.
Write the combined time as an improper fraction: hours, so the taps together fill of the tank per hour.
Let the smaller tap take hours alone, so the larger takes hours:
Dividing through by :
giving or .
Now the rejection. If then the larger tap would take hours, which is impossible. **So the smaller tap takes hours and the larger takes hours.
Check**: . Correct, and hours is indeed .
The rejection here was not about negativity of the root itself — is a positive number. It was rejected because it made another quantity in the problem negative. Always test every quantity the root generates, not just the root.
Worked example 3 — a right triangle. The altitude of a right triangle is cm less than its base. If the hypotenuse is cm, find the other two sides.
Let the base be cm, so the altitude is cm. By the Pythagoras theorem,
so or . **A length cannot be negative, so the base is cm and the altitude is cm.
Check**: . Correct.
Worked example 4 — a train's speed. A train travels km at a uniform speed. Had its speed been km/h more, it would have taken hour less. Find its speed.
Let the speed be km/h:
giving or . **A speed cannot be negative, so the train travels at km/h.
Check**: hours, and hours, which is hour less. Correct.
One pattern runs through all four. Each problem gave a relation between a quantity and its reciprocal, or between a quantity and its square, and clearing the denominators produced the quadratic. If you can write the relation, the equation writes itself — which is why the translation step deserves more care than the solving.
Worked example 1 — a boat in a stream. The speed of a boat in still water is km/h. It goes km upstream and returns downstream to the starting point in hours minutes. Find the speed of the stream.
Let the stream's speed be km/h, so the upstream speed is and the downstream speed is . The total time is hours:
Combining the fractions:
so or . **A speed cannot be negative, so the stream flows at km/h.
Check**: upstream at km/h takes hours, downstream at km/h takes hours, and the total is hours. Correct.
Notice that the numerator simplified beautifully because the terms cancelled. That happens whenever the same distance is covered both ways, and spotting it saves a page of algebra.
Worked example 2 — two taps. Two water taps together can fill a tank in hours. The larger tap takes hours less than the smaller one to fill the tank alone. Find the time each takes separately.
Write the combined time as an improper fraction: hours, so the taps together fill of the tank per hour.
Let the smaller tap take hours alone, so the larger takes hours:
Dividing through by :
giving or .
Now the rejection. If then the larger tap would take hours, which is impossible. **So the smaller tap takes hours and the larger takes hours.
Check**: . Correct, and hours is indeed .
The rejection here was not about negativity of the root itself — is a positive number. It was rejected because it made another quantity in the problem negative. Always test every quantity the root generates, not just the root.
Worked example 3 — a right triangle. The altitude of a right triangle is cm less than its base. If the hypotenuse is cm, find the other two sides.
Let the base be cm, so the altitude is cm. By the Pythagoras theorem,
so or . **A length cannot be negative, so the base is cm and the altitude is cm.
Check**: . Correct.
Worked example 4 — a train's speed. A train travels km at a uniform speed. Had its speed been km/h more, it would have taken hour less. Find its speed.
Let the speed be km/h:
giving or . **A speed cannot be negative, so the train travels at km/h.
Check**: hours, and hours, which is hour less. Correct.
One pattern runs through all four. Each problem gave a relation between a quantity and its reciprocal, or between a quantity and its square, and clearing the denominators produced the quadratic. If you can write the relation, the equation writes itself — which is why the translation step deserves more care than the solving.
Exam tip
Which steps protect the marks in a formula question?
**List , and with signs, compute on its own line, then substitute. Splitting the work in three makes each part markable even if a later step goes wrong.
- Write the equation in standard form first**, and divide by any common factor to keep the numbers small — becomes
- **Compute as a separate step, and state the nature of the roots from its sign before solving
- Remember that is positive when is negative** — for the formula begins
- Simplify the surd: , not
- Cancel a common factor from the whole fraction, numerator and denominator together
- **Check surd roots against and rather than by substituting messy expressions
- Read whether the question wants , , or — the answer may be an inequality
- When the unknown letter is the coefficient of , reject any value that makes it zero and say why
- In a word problem, test every quantity your root produces, not just the root itself
- Write "no real roots", never "no solution"
The misconception to name. A negative discriminant does not mean you made an arithmetic mistake. Some equations genuinely have no real roots**, and is one of them. A student who assumes an error and recomputes three times has lost the time the discriminant was supposed to save.
A second trap. Dividing by cancelling only part of it. Every term must be divided, giving , and the quickest guard is the sum-of-roots check: the two roots must add to , which they do, whereas a badly cancelled pair would not.
- Write the equation in standard form first**, and divide by any common factor to keep the numbers small — becomes
- **Compute as a separate step, and state the nature of the roots from its sign before solving
- Remember that is positive when is negative** — for the formula begins
- Simplify the surd: , not
- Cancel a common factor from the whole fraction, numerator and denominator together
- **Check surd roots against and rather than by substituting messy expressions
- Read whether the question wants , , or — the answer may be an inequality
- When the unknown letter is the coefficient of , reject any value that makes it zero and say why
- In a word problem, test every quantity your root produces, not just the root itself
- Write "no real roots", never "no solution"
The misconception to name. A negative discriminant does not mean you made an arithmetic mistake. Some equations genuinely have no real roots**, and is one of them. A student who assumes an error and recomputes three times has lost the time the discriminant was supposed to save.
A second trap. Dividing by cancelling only part of it. Every term must be divided, giving , and the quickest guard is the sum-of-roots check: the two roots must add to , which they do, whereas a badly cancelled pair would not.
Did you know
Why does the discriminant appear inside the formula at all?
The formula is not a rule handed down from nowhere. It is what you get if you complete the square on the general equation, and the discriminant is simply what ends up under the square root sign.
Start with and divide by :
Add and subtract the square of half the coefficient of :
And there it is. The whole question of how many real roots exist has been reduced to whether the right-hand side is positive, zero or negative — because a square on the left cannot be negative.
- **If you can take the square root two ways, giving two roots
- If the square of a quantity is zero, so the quantity is zero and there is one value
- If ** no real number has a square equal to a negative number, so there is no real root
Taking the square root of both sides finishes it:
So the formula and the discriminant test are one derivation, not two topics. Once you have seen this, the three cases stop being a list to memorise and become obvious.
It also explains the shape of the equal-root case. When the formula reduces to , which is exactly the value that makes the completed square vanish — **and it is the -coordinate of the parabola's turning point.** That is why a parabola with equal roots touches the axis rather than crossing it: the axis passes through the lowest or highest point of the curve.
One more thing the completed square gives you for free. Because is never negative, the value of the quadratic is bounded: with it has a minimum and with a maximum, both occurring at . That is how the largest area or the least cost is found in problems you will meet later, and it comes out of the same two lines of algebra that produced the formula.
Start with and divide by :
Add and subtract the square of half the coefficient of :
And there it is. The whole question of how many real roots exist has been reduced to whether the right-hand side is positive, zero or negative — because a square on the left cannot be negative.
- **If you can take the square root two ways, giving two roots
- If the square of a quantity is zero, so the quantity is zero and there is one value
- If ** no real number has a square equal to a negative number, so there is no real root
Taking the square root of both sides finishes it:
So the formula and the discriminant test are one derivation, not two topics. Once you have seen this, the three cases stop being a list to memorise and become obvious.
It also explains the shape of the equal-root case. When the formula reduces to , which is exactly the value that makes the completed square vanish — **and it is the -coordinate of the parabola's turning point.** That is why a parabola with equal roots touches the axis rather than crossing it: the axis passes through the lowest or highest point of the curve.
One more thing the completed square gives you for free. Because is never negative, the value of the quadratic is bounded: with it has a minimum and with a maximum, both occurring at . That is how the largest area or the least cost is found in problems you will meet later, and it comes out of the same two lines of algebra that produced the formula.
Exam relevance
How is the discriminant used in JEE?
This is foundation work for Class 11 Quadratic Equations and Complex Numbers, and the discriminant is one of the most reused single ideas in JEE Main and JEE Advanced algebra.
Where the discriminant leads. Class 11 keeps unchanged and adds a whole layer: the roots are rational when is a perfect square and the coefficients are rational, irrational conjugates when is positive but not a perfect square, and complex conjugates when is negative. The Class 10 phrase "no real roots" becomes "a pair of conjugate complex roots", and the discriminant is what decides which.
Where the sign of a quadratic leads. JEE asks constantly for the values of a parameter that make a quadratic expression positive for every real . The condition is together with , and it is built entirely from what you learn here. **The equal-root case, , is the boundary of that region**, which is why find- questions are the entry point to it.
Where the completed square leads. Class 11 uses it for the range of a quadratic expression and for the vertex of a parabola in Conic Sections; Class 12 uses the same turning-point value in Application of Derivatives. **The value appears in all three chapters, and it comes out of the derivation above.
Where the reciprocal word problems lead. Equations reducible to quadratics — with the unknown in a denominator, under a square root, or as an exponent — are a recurring JEE Main type. The taps and train problems here are the same manipulation with a story attached, and the habit of checking that a root does not make a denominator vanish becomes the habit of checking for extraneous solutions.
Where the rejection step leads. Competitive papers frequently produce two algebraic answers of which only one satisfies a stated restriction. Testing every quantity the root generates, not just the root, is exactly the discipline needed.
Question types to expect.** At this level: solve by formula, state the nature of the roots, find for equal roots, and a word problem. In competitive papers: sign of a quadratic over the reals, roots in a given interval, common roots of two quadratics, and equations reducible to quadratic form.
The single trap that costs marks. Using when the question asked for real roots. **"Real roots" means and gives an inequality, and a single value in place of a range scores nothing even though the algebra was right.
A second trap. Forgetting to reject the value that kills the leading coefficient. In the value satisfies but destroys the equation, and the identical structure appears in JEE questions where a parameter may reduce a quadratic to a linear equation with one root instead of two.
Board versus competitive emphasis. The CBSE paper marks the standard form, the discriminant, the substitution and the written rejection; a competitive paper marks the parameter range. The transferable habit is computing the discriminant before deciding anything** — it tells you how many answers exist, whether factorisation will work, and where the boundary of a parameter condition lies, all from one subtraction.
Where the discriminant leads. Class 11 keeps unchanged and adds a whole layer: the roots are rational when is a perfect square and the coefficients are rational, irrational conjugates when is positive but not a perfect square, and complex conjugates when is negative. The Class 10 phrase "no real roots" becomes "a pair of conjugate complex roots", and the discriminant is what decides which.
Where the sign of a quadratic leads. JEE asks constantly for the values of a parameter that make a quadratic expression positive for every real . The condition is together with , and it is built entirely from what you learn here. **The equal-root case, , is the boundary of that region**, which is why find- questions are the entry point to it.
Where the completed square leads. Class 11 uses it for the range of a quadratic expression and for the vertex of a parabola in Conic Sections; Class 12 uses the same turning-point value in Application of Derivatives. **The value appears in all three chapters, and it comes out of the derivation above.
Where the reciprocal word problems lead. Equations reducible to quadratics — with the unknown in a denominator, under a square root, or as an exponent — are a recurring JEE Main type. The taps and train problems here are the same manipulation with a story attached, and the habit of checking that a root does not make a denominator vanish becomes the habit of checking for extraneous solutions.
Where the rejection step leads. Competitive papers frequently produce two algebraic answers of which only one satisfies a stated restriction. Testing every quantity the root generates, not just the root, is exactly the discipline needed.
Question types to expect.** At this level: solve by formula, state the nature of the roots, find for equal roots, and a word problem. In competitive papers: sign of a quadratic over the reals, roots in a given interval, common roots of two quadratics, and equations reducible to quadratic form.
The single trap that costs marks. Using when the question asked for real roots. **"Real roots" means and gives an inequality, and a single value in place of a range scores nothing even though the algebra was right.
A second trap. Forgetting to reject the value that kills the leading coefficient. In the value satisfies but destroys the equation, and the identical structure appears in JEE questions where a parameter may reduce a quadratic to a linear equation with one root instead of two.
Board versus competitive emphasis. The CBSE paper marks the standard form, the discriminant, the substitution and the written rejection; a competitive paper marks the parameter range. The transferable habit is computing the discriminant before deciding anything** — it tells you how many answers exist, whether factorisation will work, and where the boundary of a parameter condition lies, all from one subtraction.
Key takeaways
What must you be able to do from this part?
One formula, one discriminant and one honest check on every root.
- The quadratic formula is , and it solves every quadratic equation
- ** is positive when is negative**, and the whole numerator sits over
- **** has and roots and ; **** has and roots and
- **** has and roots — simplify the surd and cancel the whole fraction
- **The discriminant is : positive gives real and distinct roots, zero gives real and equal roots, negative gives no real roots
- a perfect square with rational coefficients** means rational roots, so factorisation will work; positive but not a perfect square means irrational conjugate roots
- ** has and genuinely has no real roots
- has ** and the repeated root
- Graphically: cuts the axis twice, touches it once, misses it
- **For equal roots set ** — gives
- **Reject a value that makes zero** — gives , not
- **For real roots set **, whose answer is an inequality such as or
- **A boat at km/h covering km each way in hours** meets a stream of km/h
- **Taps filling a tank in hours with a -hour gap** take and hours, rejecting because it makes the other time negative
- **A right triangle with hypotenuse cm and altitude cm less than the base** has sides cm and cm
- **A km journey that would be an hour shorter at km/h more** is made at km/h
- Check surd roots against sum and product
The fastest self-test is one subtraction. Write down three quadratics of your own, predict from alone whether each will factorise, and then try — if the prediction holds every time, the discriminant is working for you rather than being one more thing to remember.
- The quadratic formula is , and it solves every quadratic equation
- ** is positive when is negative**, and the whole numerator sits over
- **** has and roots and ; **** has and roots and
- **** has and roots — simplify the surd and cancel the whole fraction
- **The discriminant is : positive gives real and distinct roots, zero gives real and equal roots, negative gives no real roots
- a perfect square with rational coefficients** means rational roots, so factorisation will work; positive but not a perfect square means irrational conjugate roots
- ** has and genuinely has no real roots
- has ** and the repeated root
- Graphically: cuts the axis twice, touches it once, misses it
- **For equal roots set ** — gives
- **Reject a value that makes zero** — gives , not
- **For real roots set **, whose answer is an inequality such as or
- **A boat at km/h covering km each way in hours** meets a stream of km/h
- **Taps filling a tank in hours with a -hour gap** take and hours, rejecting because it makes the other time negative
- **A right triangle with hypotenuse cm and altitude cm less than the base** has sides cm and cm
- **A km journey that would be an hour shorter at km/h more** is made at km/h
- Check surd roots against sum and product
The fastest self-test is one subtraction. Write down three quadratics of your own, predict from alone whether each will factorise, and then try — if the prediction holds every time, the discriminant is working for you rather than being one more thing to remember.