Square 97 in Your Head Without Writing Anything Down
Learn the three basic identities for squares and the difference of squares, see why the square of a sum works as an area picture, factorise by recognising the matching identity, and square numbers near 100 mentally.
How do you square 97 without long multiplication?
Multiplying the usual way takes three partial products and a carry. There is a faster route.
is , and the square of a difference has a fixed shape:
Three easy numbers, no carrying, and it can be done while walking.
The same shape works for , and a different one makes almost instant:
An identity is an equation true for every value of its letters, and that is what makes these tricks legal rather than lucky. holds whether and are and , or and , or anything else — so one algebraic fact serves both mental arithmetic and factorisation.
This page covers the first part of the CBSE Class 9 Mathematics chapter on algebraic identities — the three basic ones, the area picture that proves the first, using them backwards to factorise, and using them on numbers.
is , and the square of a difference has a fixed shape:
Three easy numbers, no carrying, and it can be done while walking.
The same shape works for , and a different one makes almost instant:
An identity is an equation true for every value of its letters, and that is what makes these tricks legal rather than lucky. holds whether and are and , or and , or anything else — so one algebraic fact serves both mental arithmetic and factorisation.
This page covers the first part of the CBSE Class 9 Mathematics chapter on algebraic identities — the three basic ones, the area picture that proves the first, using them backwards to factorise, and using them on numbers.
Formula
What are the three identities you have to know by heart?
Two for squaring a two-term expression, and one for a product that collapses:
The second is the first with replaced by , so there is really one squaring identity with a sign that follows the bracket. The third is the one that produces two terms instead of three, because the middle terms cancel: .
Worked example 1. Expand . Here and :
Check at , : the bracket is , and the expansion gives . Correct.
Worked example 2. Expand :
Check at : , and . Correct.
Worked example 3. Expand . Same first terms, opposite signs — the third identity:
Worked example 4 — squaring a three-term bracket is not this identity. , but needs the extended version from the next part of this chapter. Recognising which identity applies is half the work.
The error that outnumbers every other in this chapter. is not . Put in numbers: with and ,
The missing is exactly . The middle term is the whole content of the identity, and dropping it is not a small slip — it changes the answer by roughly half. The next section shows what that middle term is, as a piece of area you can point at.
The second is the first with replaced by , so there is really one squaring identity with a sign that follows the bracket. The third is the one that produces two terms instead of three, because the middle terms cancel: .
Worked example 1. Expand . Here and :
Check at , : the bracket is , and the expansion gives . Correct.
Worked example 2. Expand :
Check at : , and . Correct.
Worked example 3. Expand . Same first terms, opposite signs — the third identity:
Worked example 4 — squaring a three-term bracket is not this identity. , but needs the extended version from the next part of this chapter. Recognising which identity applies is half the work.
The error that outnumbers every other in this chapter. is not . Put in numbers: with and ,
The missing is exactly . The middle term is the whole content of the identity, and dropping it is not a small slip — it changes the answer by roughly half. The next section shows what that middle term is, as a piece of area you can point at.
Why does the square of a sum have a middle term?
**Because a square of side splits into four pieces, and two of them are the rectangles .**
Draw a square whose side is . Cut it once across and once down, at distance from the same corner. Four regions appear:
- a square of side , area
- a square of side , area
- a rectangle by , area
- another rectangle by , area
The whole square's area is , and the four pieces must add to it:
So the middle term is two rectangles, and the reason falls short is that it accounts for the two corner squares and forgets the rectangles between them.
Worked example — with numbers on the picture. Take cm and cm, a square of side cm.
- Total area:
- Big corner square:
- Small corner square:
- Two rectangles:
The pieces account for the square exactly. And , short of by the of rectangle area — the arithmetic version of the same gap.
**The picture for .** Start from a square of side and cut a strip of width off two adjacent sides. Each strip has area , so removing both takes away — but the little corner square of area has been removed twice, once by each strip, so it must be added back:
**This is where the plus sign on comes from**, and it is worth holding on to, because writing is the second most common error in the chapter. The corner was double-subtracted; the repairs it.
A geometric argument like this needs positive lengths to make sense, since a rectangle cannot have a negative side. The algebra, though, holds for every value including negatives — which is why the identity is proved by expanding the brackets and merely illustrated by the areas.
Draw a square whose side is . Cut it once across and once down, at distance from the same corner. Four regions appear:
- a square of side , area
- a square of side , area
- a rectangle by , area
- another rectangle by , area
The whole square's area is , and the four pieces must add to it:
So the middle term is two rectangles, and the reason falls short is that it accounts for the two corner squares and forgets the rectangles between them.
Worked example — with numbers on the picture. Take cm and cm, a square of side cm.
- Total area:
- Big corner square:
- Small corner square:
- Two rectangles:
The pieces account for the square exactly. And , short of by the of rectangle area — the arithmetic version of the same gap.
**The picture for .** Start from a square of side and cut a strip of width off two adjacent sides. Each strip has area , so removing both takes away — but the little corner square of area has been removed twice, once by each strip, so it must be added back:
**This is where the plus sign on comes from**, and it is worth holding on to, because writing is the second most common error in the chapter. The corner was double-subtracted; the repairs it.
A geometric argument like this needs positive lengths to make sense, since a rectangle cannot have a negative side. The algebra, though, holds for every value including negatives — which is why the identity is proved by expanding the brackets and merely illustrated by the areas.
How do you factorise by spotting which identity fits?
Read the identities right to left. An expression that matches the right-hand side can be replaced by the left, and that is factorisation.
Worked example 1 — a difference of squares. Factorise . Both terms are squares and there is a minus between them, so the third identity applies with , :
Check by expanding: . Correct.
Worked example 2 — a perfect square. Factorise . The outer terms are and ; the test is whether the middle term is . It is:
Worked example 3 — with coefficients. Factorise :
Worked example 4 — a negative middle term. Factorise . Outer terms and ; middle term should be , and it is, with a minus:
Check at : , and . Correct.
Worked example 5 — take out a common factor first. Factorise . Neither term is a perfect square as written, but
Always check for a common factor before hunting for an identity — it often converts a non-match into a match.
Worked example 6 — a difference of squares hiding in brackets. Factorise . Treat the bracket as a single object with and :
Check at : , and . Correct.
A sum of squares does not factorise this way. has no factorisation into real brackets — there is no identity for . Students who have just met the difference of squares reach for , which expands to and is a different expression. The minus sign is not decoration; it is the identity's engine, and it is the first thing to check before starting.
Worked example 1 — a difference of squares. Factorise . Both terms are squares and there is a minus between them, so the third identity applies with , :
Check by expanding: . Correct.
Worked example 2 — a perfect square. Factorise . The outer terms are and ; the test is whether the middle term is . It is:
Worked example 3 — with coefficients. Factorise :
Worked example 4 — a negative middle term. Factorise . Outer terms and ; middle term should be , and it is, with a minus:
Check at : , and . Correct.
Worked example 5 — take out a common factor first. Factorise . Neither term is a perfect square as written, but
Always check for a common factor before hunting for an identity — it often converts a non-match into a match.
Worked example 6 — a difference of squares hiding in brackets. Factorise . Treat the bracket as a single object with and :
Check at : , and . Correct.
A sum of squares does not factorise this way. has no factorisation into real brackets — there is no identity for . Students who have just met the difference of squares reach for , which expands to and is a different expression. The minus sign is not decoration; it is the identity's engine, and it is the first thing to check before starting.
How do you use identities to multiply numbers mentally?
Write each number as a round number plus or minus a small amount, then apply the matching identity. The round number does the heavy lifting and the small one stays easy.
**Worked example 1 — .** Take , :
**Worked example 2 — .** Now with a minus:
**Worked example 3 — .** The two numbers sit the same distance either side of , so the difference of squares applies:
Check the structure: the product of two numbers equally spaced around is always minus the square of the spacing.
**Worked example 4 — .**
Worked example 5 — a larger round number. Compute :
Worked example 6 — squaring near 50. Compute with , :
Worked example 7 — a product that is not a difference of squares. has spacings of and from , so it works: . But has spacings of and , unequal, so the identity does not apply — that one needs the identity from the next part of this chapter.
The identity chosen must match the structure, not the convenience. The two numbers being near is not enough for the difference of squares; they must be equally far from it on opposite sides. Checking that takes a second and saves a wrong answer that looks plausible.
One sanity check worth building in. The last digit of the answer must match the last digit of the ordinary multiplication: must end in , since , and does. It costs nothing and catches a dropped middle term immediately.
**Worked example 1 — .** Take , :
**Worked example 2 — .** Now with a minus:
**Worked example 3 — .** The two numbers sit the same distance either side of , so the difference of squares applies:
Check the structure: the product of two numbers equally spaced around is always minus the square of the spacing.
**Worked example 4 — .**
Worked example 5 — a larger round number. Compute :
Worked example 6 — squaring near 50. Compute with , :
Worked example 7 — a product that is not a difference of squares. has spacings of and from , so it works: . But has spacings of and , unequal, so the identity does not apply — that one needs the identity from the next part of this chapter.
The identity chosen must match the structure, not the convenience. The two numbers being near is not enough for the difference of squares; they must be equally far from it on opposite sides. Checking that takes a second and saves a wrong answer that looks plausible.
One sanity check worth building in. The last digit of the answer must match the last digit of the ordinary multiplication: must end in , since , and does. It costs nothing and catches a dropped middle term immediately.
Exam tip
Exam tip: name the identity and state a and b before expanding
**Write down which identity you are using and what and are.** One line — *using with , * — and the substitution becomes mechanical.
**Never write .** The middle term is the identity. With , : , not .
**And — the last term is plus. The corner square was subtracted twice, so it is added back.
Check every expansion by substituting .** It takes five seconds: at gives , and .
For factorising, test the middle term. is because . If the middle term does not match, it is not a perfect square.
Take out a common factor first: .
A sum of squares does not factorise. stays as it is — the minus sign is what makes the difference of squares work.
Treat a bracket as a single letter when it fits: .
For mental arithmetic, check the spacing is equal before using : works, does not.
And verify the last digit against the ordinary product — must end in .
For the geometric justification, draw and label the four regions; the marks are for the diagram with , and two rectangles named, not for the algebra alone.
**Never write .** The middle term is the identity. With , : , not .
**And — the last term is plus. The corner square was subtracted twice, so it is added back.
Check every expansion by substituting .** It takes five seconds: at gives , and .
For factorising, test the middle term. is because . If the middle term does not match, it is not a perfect square.
Take out a common factor first: .
A sum of squares does not factorise. stays as it is — the minus sign is what makes the difference of squares work.
Treat a bracket as a single letter when it fits: .
For mental arithmetic, check the spacing is equal before using : works, does not.
And verify the last digit against the ordinary product — must end in .
For the geometric justification, draw and label the four regions; the marks are for the diagram with , and two rectangles named, not for the algebra alone.
Did you know
Why two numbers around 100 always give a product just under 10000
Pick any two numbers that add to and multiply them.
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The products only fall, never rise, and the identity says exactly why. Any such pair is and , so
The product is minus a square, and a square is never negative. So is the largest product available, and every step away from the middle costs .
The general statement is that among all rectangles with a fixed perimeter, the square has the greatest area. A fence of m encloses as a m square, as a by rectangle, and as a by one. Same fence, different field.
This is a useful thing to know if you ever have a fixed amount of boundary and want the most land, and it explains why a plot described only by its perimeter tells you nothing definite about its area.
What makes it pleasant is that the whole result is contained in , read the other way round. The identity is usually met as a shortcut for multiplication, and it turns out to answer a question about fences — because is not just a fast way to compute , it is a statement about how the product behaves as grows.
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The products only fall, never rise, and the identity says exactly why. Any such pair is and , so
The product is minus a square, and a square is never negative. So is the largest product available, and every step away from the middle costs .
The general statement is that among all rectangles with a fixed perimeter, the square has the greatest area. A fence of m encloses as a m square, as a by rectangle, and as a by one. Same fence, different field.
This is a useful thing to know if you ever have a fixed amount of boundary and want the most land, and it explains why a plot described only by its perimeter tells you nothing definite about its area.
What makes it pleasant is that the whole result is contained in , read the other way round. The identity is usually met as a shortcut for multiplication, and it turns out to answer a question about fences — because is not just a fast way to compute , it is a statement about how the product behaves as grows.
Exam relevance
How are algebraic identities tested in JEE Main?
Because they are the manipulation tools every later algebra chapter assumes, and a student who has to expand slowly loses time on problems that are not about that at all.
This is the foundation for Class 10 Polynomials and Quadratic Equations, and then for Class 11 Mathematics Complex Numbers and Quadratic Equations and Binomial Theorem, all examined in JEE Main. The square of a sum is the case of the binomial theorem, and the coefficients seen here are the row of Pascal's triangle that the general theorem extends to and beyond — the next part of this chapter takes the cube, and Class 11 takes every power.
Completing the square is where the squaring identity earns the most later. Solving means recognising that is , which is nothing but the identity used to manufacture a perfect square. The quadratic formula itself is derived that way, and Class 11 Conic Sections uses the same move to put the equation of a circle or parabola into standard form. **A student who can see the missing has the whole method.
The difference of squares** becomes the standard factorisation move and the conjugate trick. Rationalising works by multiplying by so that the denominator collapses to — the same identity, applied to surds. In Class 11 Complex Numbers it reappears as , which is how a complex denominator is cleared.
Where the geometric picture leads. The area proof on this page is the accessible form of an argument used throughout coordinate geometry, and the fence result in the previous section is a first encounter with maxima and minima, treated properly in Class 12 Application of Derivatives.
What the questions look like. For board work, expect expand using an identity, factorise by recognising one, evaluate a numerical square or product mentally, and **justify geometrically with a labelled figure — the last one is asked in words and needs the diagram. For JEE Main, identities never appear as the question; they appear inside simplification steps, and the marks are lost to slowness rather than to ignorance.
How board and competitive emphasis differ. A board paper rewards the written steps** — naming the identity, stating and , drawing the regions. A competitive paper rewards recognising a perfect square or a difference of squares on sight, because the recognition is usually step one of five.
The single trap that costs the most marks. Assuming a sum of squares factorises. splits into and does not split at all over the real numbers. Questions deliberately place the two side by side, and the student who has learnt the identity as a pattern rather than as an equation will factorise both.
This is the foundation for Class 10 Polynomials and Quadratic Equations, and then for Class 11 Mathematics Complex Numbers and Quadratic Equations and Binomial Theorem, all examined in JEE Main. The square of a sum is the case of the binomial theorem, and the coefficients seen here are the row of Pascal's triangle that the general theorem extends to and beyond — the next part of this chapter takes the cube, and Class 11 takes every power.
Completing the square is where the squaring identity earns the most later. Solving means recognising that is , which is nothing but the identity used to manufacture a perfect square. The quadratic formula itself is derived that way, and Class 11 Conic Sections uses the same move to put the equation of a circle or parabola into standard form. **A student who can see the missing has the whole method.
The difference of squares** becomes the standard factorisation move and the conjugate trick. Rationalising works by multiplying by so that the denominator collapses to — the same identity, applied to surds. In Class 11 Complex Numbers it reappears as , which is how a complex denominator is cleared.
Where the geometric picture leads. The area proof on this page is the accessible form of an argument used throughout coordinate geometry, and the fence result in the previous section is a first encounter with maxima and minima, treated properly in Class 12 Application of Derivatives.
What the questions look like. For board work, expect expand using an identity, factorise by recognising one, evaluate a numerical square or product mentally, and **justify geometrically with a labelled figure — the last one is asked in words and needs the diagram. For JEE Main, identities never appear as the question; they appear inside simplification steps, and the marks are lost to slowness rather than to ignorance.
How board and competitive emphasis differ. A board paper rewards the written steps** — naming the identity, stating and , drawing the regions. A competitive paper rewards recognising a perfect square or a difference of squares on sight, because the recognition is usually step one of five.
The single trap that costs the most marks. Assuming a sum of squares factorises. splits into and does not split at all over the real numbers. Questions deliberately place the two side by side, and the student who has learnt the identity as a pattern rather than as an equation will factorise both.
Key takeaways
Squares, differences of squares and mental multiplication: quick revision
- An identity is an equation true for every value of its letters — which is what makes it safe to substitute anything.
- The three to know: ; ; .
- ; ; .
- ** is never .** With , : against , and the missing is .
- **In the last term is plus — the corner square was subtracted twice and must be added back.
- The area picture**: a square of side splits into , and two rectangles. With , : .
- The geometric argument needs positive lengths; the algebra holds for all values, which is why the proof is the expansion.
- Factorising is reading the identities backwards. ; .
- because — always test the middle term.
- , checked at : .
- Common factor first: .
- Treat a bracket as one letter: .
- A sum of squares does not factorise: stays as it is.
- Mental squares: ; ; ; ; .
- Mental products: ; .
- The spacings must be equal for — does not qualify.
- Check the last digit against the ordinary product: must end in .
- Two numbers adding to give , so is the largest — the same reason a fixed perimeter encloses the most area as a square.
Pick any two-digit number, square it using a round number and a small adjustment, then check it against ordinary multiplication — the second time you do it, you will stop needing the check.
- The three to know: ; ; .
- ; ; .
- ** is never .** With , : against , and the missing is .
- **In the last term is plus — the corner square was subtracted twice and must be added back.
- The area picture**: a square of side splits into , and two rectangles. With , : .
- The geometric argument needs positive lengths; the algebra holds for all values, which is why the proof is the expansion.
- Factorising is reading the identities backwards. ; .
- because — always test the middle term.
- , checked at : .
- Common factor first: .
- Treat a bracket as one letter: .
- A sum of squares does not factorise: stays as it is.
- Mental squares: ; ; ; ; .
- Mental products: ; .
- The spacings must be equal for — does not qualify.
- Check the last digit against the ordinary product: must end in .
- Two numbers adding to give , so is the largest — the same reason a fixed perimeter encloses the most area as a square.
Pick any two-digit number, square it using a round number and a small adjustment, then check it against ordinary multiplication — the second time you do it, you will stop needing the check.