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Factorising Harder Quadratic Expressions
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Factorising Harder Quadratic Expressions

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Factorising Harder Quadratic Expressions

In some quadratic expressions, the coefficient of the squared term is more than one.

Have a look at this example

1

What is the coefficient for x2x^2 in this quadratic expression?

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2

When x2x^2 has a coefficient, you factorise a quadratic expression into the form (ax+b)(cx+d)(ax+b)(cx+d)

First you want to find out what axax and cxcx might be.

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3

axax multiplied by cxcx have to give 4x24x^2. Which 2 of these options equal 4x24x^2?

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You can select multiple answers

4

So there are two options for what axax and cxcx are in the factorised expression

You will come back to finding out which option is the correct one in a minute.

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5

Now you look at bb and dd. When they are multiplied, they have to equal 99. Which 2 of these options fulfil that?

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You can select multiple answers

6

So there are also two options for what bb and dd are in the factorised expression

That means there are 4 options for what the factorised expression might be!

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7

The correct factorised expression results in 4x2+12x+94x^2 + 12x +9 when you expand the brackets. Which one of the options is that?

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8

So option C is the correct factorisation of 4x2+12x+94x^2+12x+9

The other options don't give you the correct expression when you expand the brackets.

Option A gives you 4x2+15x+94x^2+15x+9

Option B gives you 4x2+13x+94x^2+13x+9

Option D gives you 4x2+20x+94x^2+20x+9

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Recap! Sometimes x2x^2 in a quadratic expression will have a coefficient in front of it. There are 3 steps to factorising the expression then.

1

First you find options for axax and cxcx

If you multiply axax and cxcx is should equal the first term in the original expression.

Here the first term is 6x26x^2, so the options are 2x×3x2x \times 3x and 6x×x6x \times x

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2

Then you look at options for bb and dd

If you multiply bb and dd it should equal the last term in the original expression.

Here the last term is 88, so the options are 4×24 \times 2 and 8×18 \times 1

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3

Finally, you can now expand all the options to find out which option is correct

In this example, that is the option where the middle term is 16x16x once you have expanded the brackets and simplified the expression.

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What is the correct factorisation of 9x2+11x+29x^2+11x+2?

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Factorise 2x2+10x+82x^2+10x+8 into the form (ax+b)(cx+d)(ax + b)(cx+d)

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Now take a look at this example

1

Here one of the terms is negative

You need to factorise so that you will end up with 12x-12x as the middle term.

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2

True or false? If axax, cxcx, bb and dd are all positive, all terms will also be positive once you have expanded the brackets.

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3

So to get to a negative term when you expand the brackets, at least one of the terms in the factorised expression has to be _____________.

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4

First you find the options for axax and cxcx

They are either 4x×x4x \times x or 2x×2x2x \times 2x

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5

Now, the options for bb and dd have to make +9+9 when multiplied. Pick all the options below that equal +9+9

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You can select multiple answers

6

So there are actually 4 options for bb and dd

Two pairs of positives and two pairs of negatives.

You could also write the options as ±3×±3\pm 3 \times \pm3 and ±9×±1\pm9 \times \pm1 to save space.

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7

You can exclude two of these pairs as correct options for bb and dd. Which ones?

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8

So now you are down to two options for bb and dd. That gives you 4 options for factorised expressions. Which one is the correct one?

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9

So when one of the terms in the original expression is negative, at least one of the terms in the factorised expression has to also be negative

Here both bb and dd are negative, because when they are multiplied they have to give the positive number +9+9

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What is the correct factorisation of 6x2+19x76x^2+19x-7?

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Factorise 5x222x+85x^2-22x+8 into the form (ax+b)(cx+d)(ax + b)(cx+d)

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Summary! There are 3 steps to factorising these harder quadratic expressions where there is a coefficient in front of x2x^2

1

First you find options for axax and cxcx

If you multiply axax and cxcx is should equal the first term in the original expression.

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2

Then you look at options for bb and dd

If you multiply bb and dd it should equal the last term in the original expression.

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3

Finally, you can now expand all the options to find out which option is correct

In this example, that is the option where the middle term is 22x-22x once you have expanded the brackets and simplified the expression.

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4

When one of the terms in the original expression is negative, at least one of the terms in the factorised expression has to also be negative

If you only have positive terms, all terms will be positive when you expand the brackets.

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