Add a Term and Take It Away Again to Reveal a Hidden Factor
Take out the highest common factor, split four or six terms into workable groups, spot a difference of two squares even when it is disguised, and manufacture one by completing the square.
How can adding something you then subtract make an expression easier?
Look at and try to factorise it. There is no common factor, only two terms, and it is not a difference of squares — it is a sum. It looks finished.
Now do something that seems pointless. Add and subtract :
Nothing has changed in value, because the two extra terms cancel. But the first three terms are now a perfect square:
and that is a difference of two squares, which factorises at once:
**Check it at **: the original is , and the factors give . Correct.
So the trick was to add a term in order to create a square, and subtract it again to keep the value. That technique is called completing the square, and it is the last and most interesting of the four methods on this page.
Factorising is the reverse of expanding. Last chapter you multiplied brackets out; here you put them back. And there are only four tools:
- Take out a common factor
- Group the terms and take out a common factor from each group
- Recognise a difference of two squares
- Complete the square to manufacture one
This page covers the first part of the ICSE Class 9 Mathematics chapter on factorisation: the highest common factor, grouping, the difference of two squares, and completing the square.
Now do something that seems pointless. Add and subtract :
Nothing has changed in value, because the two extra terms cancel. But the first three terms are now a perfect square:
and that is a difference of two squares, which factorises at once:
**Check it at **: the original is , and the factors give . Correct.
So the trick was to add a term in order to create a square, and subtract it again to keep the value. That technique is called completing the square, and it is the last and most interesting of the four methods on this page.
Factorising is the reverse of expanding. Last chapter you multiplied brackets out; here you put them back. And there are only four tools:
- Take out a common factor
- Group the terms and take out a common factor from each group
- Recognise a difference of two squares
- Complete the square to manufacture one
This page covers the first part of the ICSE Class 9 Mathematics chapter on factorisation: the highest common factor, grouping, the difference of two squares, and completing the square.
How do you take out the highest common factor?
Find the largest number and the lowest power of each letter that divides every term, and take all of it out at once.
The HCF of the terms has three parts: the HCF of the coefficients, and for each letter the lowest power in which it appears in every term.
Worked example 1. Factorise .
- HCF of , and is
- Lowest power of across the three terms is
- Lowest power of across the three terms is
So the HCF is . Divide each term by it:
Always check by expanding back. ; ; . All three match.
Worked example 2. Factorise .
HCF of and is ; lowest power of is ; lowest power of is . So the HCF is :
Now the two errors this step invites.
Do not stop at a factor that is not the highest. Writing is correct but incomplete, because , and still share a factor of . If the bracket's terms still have a common factor, you have not finished.
**And do not lose a term that becomes .** Factorising gives , not — the second term divided by leaves , and that must be written. A term that vanishes from the bracket is the commonest single mistake in this method, and expanding back catches it instantly.
Taking out the HCF is always the first step. Even when another method is needed afterwards, doing this first makes every later step smaller — and the last section of the next part of this chapter is entirely about combining methods in that order.
The HCF of the terms has three parts: the HCF of the coefficients, and for each letter the lowest power in which it appears in every term.
Worked example 1. Factorise .
- HCF of , and is
- Lowest power of across the three terms is
- Lowest power of across the three terms is
So the HCF is . Divide each term by it:
Always check by expanding back. ; ; . All three match.
Worked example 2. Factorise .
HCF of and is ; lowest power of is ; lowest power of is . So the HCF is :
Now the two errors this step invites.
Do not stop at a factor that is not the highest. Writing is correct but incomplete, because , and still share a factor of . If the bracket's terms still have a common factor, you have not finished.
**And do not lose a term that becomes .** Factorising gives , not — the second term divided by leaves , and that must be written. A term that vanishes from the bracket is the commonest single mistake in this method, and expanding back catches it instantly.
Taking out the HCF is always the first step. Even when another method is needed afterwards, doing this first makes every later step smaller — and the last section of the next part of this chapter is entirely about combining methods in that order.
How do you factorise four or six terms by grouping?
Split the terms into groups that each have a common factor, and hope the leftover brackets match.
The method works when, after taking a factor out of each group, the same bracket is left in every group — because that bracket can then itself be taken out.
Worked example 1 — the basic pattern. Factorise .
Group the first two and the last two:
Both groups now contain , so take it out:
Check by expanding: . All four terms present.
Worked example 2. Factorise .
Check: . Agreed.
Worked example 3 — six terms. Factorise .
Group the first three and the last three:
Worked example 4 — when the terms need rearranging first. Factorise .
The first two terms share and the last two share :
Check by expanding: . All four terms present.
Now the judgement this method requires, which is where students get stuck. The grouping is not always the obvious one. In example 4, grouping the first and third terms together instead would give and nothing useful.
The rule is to look for two terms that share a factor, and then check whether the remaining two share a factor that leaves the SAME bracket. If they do not, try a different pairing before concluding that grouping will not work.
And the sign needs watching. To factorise , the first pair gives and the second pair gives — note the minus taken out with the , which is what makes the brackets match. Writing is the same thing and looks different, which is why taking the minus out deliberately is the safer habit.
A useful sanity check on any grouping. A four-term expression that factorises this way gives two brackets of two terms each, and multiplying them back gives exactly four terms. If your expansion produces more or fewer than four, the grouping was wrong — and that check costs one line.
The method works when, after taking a factor out of each group, the same bracket is left in every group — because that bracket can then itself be taken out.
Worked example 1 — the basic pattern. Factorise .
Group the first two and the last two:
Both groups now contain , so take it out:
Check by expanding: . All four terms present.
Worked example 2. Factorise .
Check: . Agreed.
Worked example 3 — six terms. Factorise .
Group the first three and the last three:
Worked example 4 — when the terms need rearranging first. Factorise .
The first two terms share and the last two share :
Check by expanding: . All four terms present.
Now the judgement this method requires, which is where students get stuck. The grouping is not always the obvious one. In example 4, grouping the first and third terms together instead would give and nothing useful.
The rule is to look for two terms that share a factor, and then check whether the remaining two share a factor that leaves the SAME bracket. If they do not, try a different pairing before concluding that grouping will not work.
And the sign needs watching. To factorise , the first pair gives and the second pair gives — note the minus taken out with the , which is what makes the brackets match. Writing is the same thing and looks different, which is why taking the minus out deliberately is the safer habit.
A useful sanity check on any grouping. A four-term expression that factorises this way gives two brackets of two terms each, and multiplying them back gives exactly four terms. If your expansion produces more or fewer than four, the grouping was wrong — and that check costs one line.
Formula
How do you spot a difference of two squares when it is disguised?
Write the expression as something squared minus something else squared, then apply the identity.
The identity needs two terms, a minus between them, and both terms perfect squares. A sum of two squares does not factorise this way — which is exactly why needed the trick in the last section.
Worked example 1 — straightforward. Factorise .
Worked example 2 — numerical. Evaluate .
Check by squaring: . Agreed — and the identity turned two four-digit squares into a one-line multiplication.
Worked example 3 — grouping needed first. Factorise .
The first two terms are a difference of squares and the last two share a factor:
Now both parts contain :
**Check at , **: the original is , and the factors give . Correct.
Worked example 4 — squares of brackets. Factorise .
Treat each bracket as a single quantity:
Now simplify each factor:
**Check at , **: the original is , and the factors give . Correct.
Notice what made example 4 work. The identity was applied with and standing for whole brackets rather than single letters, and only afterwards were the results simplified.
That is the general habit worth building: an identity's letters can stand for anything. does not require and to be single symbols — they can be , or , or . Recognising the shape matters far more than the letters used to write it, and that is what lets the same four identities handle expressions of any size.
The identity needs two terms, a minus between them, and both terms perfect squares. A sum of two squares does not factorise this way — which is exactly why needed the trick in the last section.
Worked example 1 — straightforward. Factorise .
Worked example 2 — numerical. Evaluate .
Check by squaring: . Agreed — and the identity turned two four-digit squares into a one-line multiplication.
Worked example 3 — grouping needed first. Factorise .
The first two terms are a difference of squares and the last two share a factor:
Now both parts contain :
**Check at , **: the original is , and the factors give . Correct.
Worked example 4 — squares of brackets. Factorise .
Treat each bracket as a single quantity:
Now simplify each factor:
**Check at , **: the original is , and the factors give . Correct.
Notice what made example 4 work. The identity was applied with and standing for whole brackets rather than single letters, and only afterwards were the results simplified.
That is the general habit worth building: an identity's letters can stand for anything. does not require and to be single symbols — they can be , or , or . Recognising the shape matters far more than the letters used to write it, and that is what lets the same four identities handle expressions of any size.
How do you complete the square to create a difference of two squares?
Add the term that would make a perfect square, and subtract the same term so that nothing changes.
This is the method for expressions that look unfactorisable — typically a sum of two fourth powers, or three terms that are nearly a square.
The pattern to aim for. You want the expression written as . Since , you look at what you have and ask what is missing from that shape.
Worked example 1. Factorise .
The first and last terms are and . For the middle term would need to be , and we only have . So add and subtract :
**Check at , **: the original is , and the factors give . Correct.
Worked example 2. Factorise .
The two terms are and . For the middle term would need to be , and there is none. So add and subtract :
**Check at **: , and . Correct.
Worked example 3. Factorise .
Here the terms are and , so the missing middle term is :
**Check at **: , and . Correct.
Now the method stated as a procedure, because all three examples followed the same three steps.
- Identify the two terms that are already perfect squares — call their roots and
- Work out what the middle term of would be, namely , and see how much of it you already have
- Add and subtract the shortfall. The first three terms are then , and what you subtracted is the second square
And the essential condition, which is easy to miss. The amount you subtract must itself be a perfect square, or the difference-of-squares identity cannot be applied.
In example 1 the shortfall was . In examples 2 and 3 it was . Every one was a square — and when the shortfall is not a square, this method does not work and the expression genuinely will not factorise by these tools.
So completing the square is not a universal escape, and the check on whether it will work comes before the attempt rather than after it.
This is the method for expressions that look unfactorisable — typically a sum of two fourth powers, or three terms that are nearly a square.
The pattern to aim for. You want the expression written as . Since , you look at what you have and ask what is missing from that shape.
Worked example 1. Factorise .
The first and last terms are and . For the middle term would need to be , and we only have . So add and subtract :
**Check at , **: the original is , and the factors give . Correct.
Worked example 2. Factorise .
The two terms are and . For the middle term would need to be , and there is none. So add and subtract :
**Check at **: , and . Correct.
Worked example 3. Factorise .
Here the terms are and , so the missing middle term is :
**Check at **: , and . Correct.
Now the method stated as a procedure, because all three examples followed the same three steps.
- Identify the two terms that are already perfect squares — call their roots and
- Work out what the middle term of would be, namely , and see how much of it you already have
- Add and subtract the shortfall. The first three terms are then , and what you subtracted is the second square
And the essential condition, which is easy to miss. The amount you subtract must itself be a perfect square, or the difference-of-squares identity cannot be applied.
In example 1 the shortfall was . In examples 2 and 3 it was . Every one was a square — and when the shortfall is not a square, this method does not work and the expression genuinely will not factorise by these tools.
So completing the square is not a universal escape, and the check on whether it will work comes before the attempt rather than after it.
Exam tip
Exam tip: take out the HCF first and expand back to check
Always look for a common factor first, whatever else the expression needs. It makes every later step smaller.
The HCF is the HCF of the coefficients together with the LOWEST power of each letter present in every term.
If the bracket's terms still share a factor, you have not finished. is incomplete — the still share a .
**Never lose a term that becomes .** , not .
Expand every answer back. It takes one line and catches a lost term, a wrong sign or an incomplete HCF.
In grouping, look for two terms with a common factor and check that the other two leave the SAME bracket. If not, try a different pairing before giving up.
Take a minus out deliberately when it makes the brackets match — .
A four-term grouping gives two brackets of two terms, so expanding back must give exactly four terms.
** needs two terms, a MINUS, and both terms perfect squares. A sum of two squares does not factorise this way.
The letters in an identity can stand for whole brackets** — in they are and .
Simplify each factor after applying the identity, and watch the signs when removing the second bracket.
**To complete the square, find the two perfect squares, work out what should be, and add and subtract the shortfall.
Check that the shortfall is itself a perfect square — if it is not, the method will not work.
And verify every factorisation numerically** with a small value such as or . One substitution confirms the whole answer.
The HCF is the HCF of the coefficients together with the LOWEST power of each letter present in every term.
If the bracket's terms still share a factor, you have not finished. is incomplete — the still share a .
**Never lose a term that becomes .** , not .
Expand every answer back. It takes one line and catches a lost term, a wrong sign or an incomplete HCF.
In grouping, look for two terms with a common factor and check that the other two leave the SAME bracket. If not, try a different pairing before giving up.
Take a minus out deliberately when it makes the brackets match — .
A four-term grouping gives two brackets of two terms, so expanding back must give exactly four terms.
** needs two terms, a MINUS, and both terms perfect squares. A sum of two squares does not factorise this way.
The letters in an identity can stand for whole brackets** — in they are and .
Simplify each factor after applying the identity, and watch the signs when removing the second bracket.
**To complete the square, find the two perfect squares, work out what should be, and add and subtract the shortfall.
Check that the shortfall is itself a perfect square — if it is not, the method will not work.
And verify every factorisation numerically** with a small value such as or . One substitution confirms the whole answer.
Did you know
Why adding zero is one of the most useful moves in algebra
Adding and then subtracting changes nothing. The net effect is to add zero, and adding zero is the most obviously pointless operation available.
And yet it is what makes factorise.
The reason is that an expression's value and its appearance are different things. Adding zero cannot change the value, and it can completely change the appearance — and factorising is entirely a question of appearance. You are looking for a shape you recognise, and sometimes the shape is one step away from what is written.
The same move, under different names, runs right through mathematics.
In this chapter it is completing the square, used to manufacture a difference of squares. In the quadratic formula it is the same move, used to force into the form so that a square root can be taken. In coordinate geometry it turns into , so that a circle's centre and radius can be read off.
Its close cousin is multiplying by one, which is just as pointless and just as useful. Multiplying by is multiplying by one, and it is what rationalises a denominator. Multiplying by is multiplying by one, and it is what clears a surd out of a limit.
So two of the most powerful techniques available to a student consist of doing something that has no effect.
What they have in common is that they add structure without adding content. The value is untouched; the form becomes something you have an identity for. And that is worth recognising as a general strategy rather than three separate tricks, because the next time an expression looks stuck, the question to ask is not "what can I change?" but "what can I add that is really zero, or multiply by that is really one?"
And yet it is what makes factorise.
The reason is that an expression's value and its appearance are different things. Adding zero cannot change the value, and it can completely change the appearance — and factorising is entirely a question of appearance. You are looking for a shape you recognise, and sometimes the shape is one step away from what is written.
The same move, under different names, runs right through mathematics.
In this chapter it is completing the square, used to manufacture a difference of squares. In the quadratic formula it is the same move, used to force into the form so that a square root can be taken. In coordinate geometry it turns into , so that a circle's centre and radius can be read off.
Its close cousin is multiplying by one, which is just as pointless and just as useful. Multiplying by is multiplying by one, and it is what rationalises a denominator. Multiplying by is multiplying by one, and it is what clears a surd out of a limit.
So two of the most powerful techniques available to a student consist of doing something that has no effect.
What they have in common is that they add structure without adding content. The value is untouched; the form becomes something you have an identity for. And that is worth recognising as a general strategy rather than three separate tricks, because the next time an expression looks stuck, the question to ask is not "what can I change?" but "what can I add that is really zero, or multiply by that is really one?"
Exam relevance
Why does JEE Main keep using completing the square?
Because it is the standard way to make a quadratic yield to a square root, and it appears in three different Class 11 chapters for that reason.
This is the foundation for Class 11 Mathematics Complex Numbers and Quadratic Equations, Straight Lines and Conic Sections, examined in JEE Main. Completing the square is how the quadratic formula is derived: is rewritten as , and the root is then taken. **The discriminant that decides the nature of the roots comes out of exactly that step, and questions on real, equal or complex roots depend on it.
Conic sections are read off completed squares.** An equation such as becomes , giving the centre and radius immediately. Questions on the centre, radius, vertex or focus of a conic given in general form are standard in JEE Main, and the first step is always this one — with a three-term square where two variables are involved.
The difference of two squares is used for factorising higher polynomials. Class 11 Complex Numbers factorises over the reals exactly as this page does, and then over the complex numbers as . **The identity with imaginary is how arises, which is one of the first results of that chapter.
Grouping becomes the factor theorem. Class 11 and 12 factorise cubics and quartics by finding one root and dividing out, and grouping is the elementary form of the same reasoning — spotting shared structure. The habit of checking an answer by expanding back is what a division check does formally.
Factorisation is the tool for limits and integrals.** Class 11 Limits evaluates by factorising the numerator as and cancelling; Class 12 Integrals uses partial fractions, which require the denominator to be factorised first. A limit or an integral that cannot be started is very often a factorisation that has not been spotted, and the difference of two squares is the commonest one needed.
Symmetric expressions and the HCF appear in polynomial questions. Taking out the HCF is the first step in simplifying any rational expression, and Class 11 uses it throughout algebra and in sequences where a common factor is removed from a series.
What the questions look like. For board work, expect factorise by taking out the HCF, factorise a four-term or six-term expression by grouping, factorise a difference of two squares including disguised cases, and factorise an expression by completing the square. Every answer should be verified by expanding. For JEE Main, expect the discriminant and nature of roots, conic sections in general form, factorisation inside limits and partial fractions.
How board and competitive emphasis differ. A board paper rewards the fully factorised answer with the method visible. A competitive paper never asks you to factorise on its own — it asks for a centre, a root, a limit or an integral that cannot be reached without factorising first.
The single trap that costs the most marks. Trying to apply to a sum of two squares. does not factorise over the real numbers, and no amount of rearranging will make it. What can be done is to complete the square on a sum of two fourth powers — as shows — because there the added middle term creates a genuine difference. The defence is to check three conditions before reaching for the identity: two terms, a minus sign, and both terms perfect squares. If the sign is a plus, the identity does not apply, and completing the square is the only route worth trying.
This is the foundation for Class 11 Mathematics Complex Numbers and Quadratic Equations, Straight Lines and Conic Sections, examined in JEE Main. Completing the square is how the quadratic formula is derived: is rewritten as , and the root is then taken. **The discriminant that decides the nature of the roots comes out of exactly that step, and questions on real, equal or complex roots depend on it.
Conic sections are read off completed squares.** An equation such as becomes , giving the centre and radius immediately. Questions on the centre, radius, vertex or focus of a conic given in general form are standard in JEE Main, and the first step is always this one — with a three-term square where two variables are involved.
The difference of two squares is used for factorising higher polynomials. Class 11 Complex Numbers factorises over the reals exactly as this page does, and then over the complex numbers as . **The identity with imaginary is how arises, which is one of the first results of that chapter.
Grouping becomes the factor theorem. Class 11 and 12 factorise cubics and quartics by finding one root and dividing out, and grouping is the elementary form of the same reasoning — spotting shared structure. The habit of checking an answer by expanding back is what a division check does formally.
Factorisation is the tool for limits and integrals.** Class 11 Limits evaluates by factorising the numerator as and cancelling; Class 12 Integrals uses partial fractions, which require the denominator to be factorised first. A limit or an integral that cannot be started is very often a factorisation that has not been spotted, and the difference of two squares is the commonest one needed.
Symmetric expressions and the HCF appear in polynomial questions. Taking out the HCF is the first step in simplifying any rational expression, and Class 11 uses it throughout algebra and in sequences where a common factor is removed from a series.
What the questions look like. For board work, expect factorise by taking out the HCF, factorise a four-term or six-term expression by grouping, factorise a difference of two squares including disguised cases, and factorise an expression by completing the square. Every answer should be verified by expanding. For JEE Main, expect the discriminant and nature of roots, conic sections in general form, factorisation inside limits and partial fractions.
How board and competitive emphasis differ. A board paper rewards the fully factorised answer with the method visible. A competitive paper never asks you to factorise on its own — it asks for a centre, a root, a limit or an integral that cannot be reached without factorising first.
The single trap that costs the most marks. Trying to apply to a sum of two squares. does not factorise over the real numbers, and no amount of rearranging will make it. What can be done is to complete the square on a sum of two fourth powers — as shows — because there the added middle term creates a genuine difference. The defence is to check three conditions before reaching for the identity: two terms, a minus sign, and both terms perfect squares. If the sign is a plus, the identity does not apply, and completing the square is the only route worth trying.
Key takeaways
Factorisation by HCF, grouping, squares and completion: quick revision
- Factorisation is the reverse of expansion, and there are four tools: HCF, grouping, difference of two squares, and completing the square.
- The HCF is the HCF of the coefficients together with the lowest power of each letter appearing in every term.
- **; .
- If the bracket's terms still share a factor, you have not taken out the highest one.
- Never lose a term that becomes **: .
- Grouping works when each group leaves the same bracket. .
- **.
- Six terms**: .
- Rearranged: .
- Take a minus out when it makes the brackets match: .
- ** — needs two terms, a minus, and both terms perfect squares. A sum of two squares does not factorise this way.
- ; .
- With grouping first**: .
- With brackets as the letters: .
- An identity's letters can stand for whole brackets — recognise the shape, not the symbols.
- Completing the square: identify the two perfect squares with roots and , work out what should be, and add and subtract the shortfall.
- **.
- .
- .
- The shortfall you subtract must itself be a perfect square, or the identity cannot be applied — so the method is not universal.
- Verify every factorisation by expanding back, or by substituting a small value** such as or .
Factorise by completing the square, then check your two factors at — if the product comes to , you have it right.
- The HCF is the HCF of the coefficients together with the lowest power of each letter appearing in every term.
- **; .
- If the bracket's terms still share a factor, you have not taken out the highest one.
- Never lose a term that becomes **: .
- Grouping works when each group leaves the same bracket. .
- **.
- Six terms**: .
- Rearranged: .
- Take a minus out when it makes the brackets match: .
- ** — needs two terms, a minus, and both terms perfect squares. A sum of two squares does not factorise this way.
- ; .
- With grouping first**: .
- With brackets as the letters: .
- An identity's letters can stand for whole brackets — recognise the shape, not the symbols.
- Completing the square: identify the two perfect squares with roots and , work out what should be, and add and subtract the shortfall.
- **.
- .
- .
- The shortfall you subtract must itself be a perfect square, or the identity cannot be applied — so the method is not universal.
- Verify every factorisation by expanding back, or by substituting a small value** such as or .
Factorise by completing the square, then check your two factors at — if the product comes to , you have it right.