13. Factorising Quadratics (a>1)

Question 1

LEVEL 6

Fully factorise the following quadratic:

Select the correct answer from the options below.

A: (2x + 2)(x + 1)

B: (2x + 1)(x + 2)

C: (2x- 1)(x- 2)

D: (2x- 2)(x + 1)

CORRECT ANSWER:  B: (2x + 1)(x + 2)

WORKED SOLUTION:

Firstly, multiply the coefficient of the x^2 term by the constant term,

so 2 \times 2 = 4.

So, we want two numbers which multiply to make 4 and which add to make 5.

The pair (1, 4) satisfies these criteria. This pair then tells us to re-write our quadratic as

2x^2 + x + 4x + 2

Then, we treat the first two terms and the last two terms

as if they were separate expressions and factorise them as such. So, we get

2x^2 + x + 4x + 2
= x(2x + 1) + 2(2x + 1)

Because (2x + 1) is a factor of both our terms, we can factorise it out.

Note: If both your brackets aren’t the same then something has gone wrong with the method.

x(2x + 1) + 2(2x + 1)
= (2x + 1)(x + 2).

This is the complete factorisation of the quadratic.

Question 2

LEVEL 6

Fully factorise the following quadratic:

Select the correct answer from the options below.

A: (3x + 1)(x - 1)

B: (3x + 1)(x - 2)

C: (3x + 2)(x - 1)

D: (3x - 2)(x + 1)

CORRECT ANSWER:  C: (3x + 2)(x - 1)

WORKED SOLUTION:

Firstly, multiply the coefficient of the x^2 term by the constant term, so

3 \times -2 = -6.

So, we want two numbers which multiply to make -6 and add to make -1. The pair (2, -3) work.

This pair then tells us how to split up our x term, i.e. we rewrite our quadratic as

3x^2 + 2x - 3x - 2

Then, we treat the first two terms and the last two terms

as if they were separate expressions and factorise them as such. So, we get

3x^2 + 2x - 3x - 2
= x(3x + 2) - 1(3x + 2)

Now, because (3x + 2) is a factor of both the terms, we can factorise it out.

Note: If what’s inside both your brackets aren’t the same, then something has gone wrong with the method

Finally, we get

x(3x + 2) - 1(3x + 2)
= (3x + 2)(x - 1)

Question 3

LEVEL 6

Fully factorise the following quadratic:

Select the correct answer from the options below.

A: (4z + 3)(z - 5)

B: (2z + 3)(2z - 5)

C: (2z - 3)(2z + 5)

D: (4z - 3)(z + 5)

CORRECT ANSWER:    B: (2z + 3)(2z - 5)

 WORKED SOLUTION:

Firstly, multiply the coefficient of the z^2 term by the constant term, so

4 \times -15 = -60.

So, we want two numbers which multiply to make -60 and add to make -4. The pair (-10, 6) work.

This pair then tells us how to split up our z term, i.e. we rewrite our quadratic as

Then, we treat the first two terms and the last two terms

as if they were separate expressions and factorise them as such. So, we get

4z^2 - 10z + 6z - 15 = 2z(2z - 5) + 3(2z - 5)

Because (2z - 5) is a factor of both the terms, we can factorise it out.

Note: If both your brackets aren’t the same, then something has gone wrong with the method

Finally, we get

2z(2z - 5) + 3(2z - 5) = (2z - 5)(2z + 3)

This is the complete factorisation of the quadratic.

Question 4

LEVEL 6

Fully factorise the following quadratic:

Select the correct answer from the options below.

A: (3a - 5)^2

B: (3a + 5)(3a - 5)

C: (a - 5)(9a + 5)

D: (a + 5)(9a - 5)

CORRECT ANSWER:    B: (3a + 5)(3a - 5)

WORKED SOLUTION:

This is a case of the difference of two squares.

Recognising quadratics of this type makes factorising them a whole lot easier.

By doing so, and observing that

\sqrt{9} = 3 and \sqrt{25} = 5,

we get immediately that the factorisation of 9a^2 - 25 is

Question 5

LEVEL 6

Fully factorise the following quadratic:

Select the correct answer from the options below.

A: (7x - 1)(x + 6)

B: (7x + 6)(x - 1)

C: (7x + 2)(x - 3)

D: (7x + 3)(x - 2)

CORRECT ANSWER:    B: (7x + 6)(x - 1) 

WORKED SOLUTION:

Firstly, multiply the coefficient of the x^2 term by the constant term, so

7 \times -6 = -42. So, we want two numbers which multiply to make -42 and add to make -1.

The pair (-7, 6) satisfies these criteria.

This pair then tells us how to split up our x term, i.e. we rewrite our quadratic as

Then, we treat the first two terms and the last two terms

as if they were separate expressions and factorise them as such. So, we get

7x^2 - 7x + 6x - 6 = 7x(x - 1) + 6(x - 1)

Because (x - 1) is a factor of both the terms, we can factorise it out.

Note: If both your brackets aren’t the same, then something has gone wrong with the method

Finally, we get

7x(x - 1) + 6(x - 1) = (x - 1)(7x + 6)

This is the complete factorisation of the quadratic.