Factorising x² + bx + c When c Is Negative

How to factorise a quadratic such as $x^2 + 5x - 6$ or $x^2 - 3x - 10$ : why the two numbers have opposite signs, which one is negative, and how to decide the signs before you start.

Before you start

In lesson 3.9 you factorised $x^2 + bx + c$ by finding two numbers whose product is $c$ and whose sum is $b$ : $x^2 + 7x + 12 = (x + 3)(x + 4)\text{.}$ When $c$ is positive the two numbers have the same sign.

This lesson is about a negative $c$ , as in $x^2 + 5x - 6$ .

Why the Two Numbers Have Opposite Signs

The two numbers multiply to give $c$ . A positive times a positive is positive, and so is a negative times a negative. The only way to get a negative product is one positive number and one negative number.

When one number is positive and the other negative, their sum is the difference of their sizes, and it takes the sign of the larger one. For example, $6 + (-1) = 5$ and $(-6) + 1 = -5$ .

Factorising $x^2 + bx + c$ when $c$ is negative

The two numbers have opposite signs.

Look for a factor pair of $c$ (ignoring its sign) whose difference is the size of $b$ .

The larger number takes the sign of $b$ ; the smaller takes the other sign.

Check by expanding.

When $b$ Is Positive

Example 1: Every pair, with its signs

Factorise $x^2 + 5x - 6$ .

Solution

We need product $-6$ and sum $5$ . The pairs with product $-6$ :

Pair	Product	Sum
$1, -6$	$-6$	$-5$
$-1, 6$	$-6$	$5$ ✓
$2, -3$	$-6$	$-1$
$-2, 3$	$-6$	$1$

×	$x$	$+6$
$x$	$x^2$	$+6x$
$-1$	$-x$	$-6$

$x^2 + 6x - x - 6 = x^2 + 5x - 6$

$x^2 + 5x - 6 = (x - 1)(x + 6)$

Example 2: Using the difference

Factorise $x^2 + 4x - 12$ .

Solution

Pairs of $12$ and their differences: $1, 12$ ( $11$ ); $2, 6$ ( $4$ ) ✓; $3, 4$ ( $1$ ).

$b = +4$ is positive, so the larger number, $6$ , is positive and $2$ is negative: $+6$ and $-2$ .

Check: $(x + 6)(x - 2) = x^2 - 2x + 6x - 12 = x^2 + 4x - 12\text{.}$ ✓

$x^2 + 4x - 12 = (x + 6)(x - 2)$

When $b$ Is Negative

Example 3: The larger number negative

Factorise (a) $x^2 - 3x - 10$ ; (b) $x^2 - 2x - 8$ .

Solution

(a) Pairs of $10$ : $1, 10$ (difference $9$ ); $2, 5$ (difference $3$ ) ✓.

$b = -3$ is negative, so the larger number is negative: $-5$ and $+2$ . Check: $(-5) \times 2 = -10$ and $-5 + 2 = -3$ . ✓

(b) Pairs of $8$ : $1, 8$ ( $7$ ); $2, 4$ ( $2$ ) ✓. The larger is negative: $-4$ and $+2$ .

(a) $(x - 5)(x + 2)$ ; (b) $(x - 4)(x + 2)$

Example 4: When $b$ is $1$ or $-1$

Factorise (a) $x^2 - x - 12$ ; (b) $x^2 + x - 20$ .

Solution

(a) $b = -1$ : we need a pair of $12$ with difference $1$ , which is $3, 4$ . The larger is negative: $-4$ and $+3$ .

(b) $b = +1$ : a pair of $20$ with difference $1$ is $4, 5$ . The larger is positive: $+5$ and $-4$ .

(a) $(x - 4)(x + 3)$ ; (b) $(x + 5)(x - 4)$

Example 5: Many pairs

Factorise (a) $x^2 - 2x - 48$ ; (b) $x^2 + 4x - 96$ .

Solution

(a) Pairs of $48$ and differences: $1, 48$ ( $47$ ); $2, 24$ ( $22$ ); $3, 16$ ( $13$ ); $4, 12$ ( $8$ ); $6, 8$ ( $2$ ) ✓. The larger is negative: $-8$ and $+6$ .

(b) Pairs of $96$ : $1, 96$ ( $95$ ); $2, 48$ ( $46$ ); $3, 32$ ( $29$ ); $4, 24$ ( $20$ ); $6, 16$ ( $10$ ); $8, 12$ ( $4$ ) ✓. The larger is positive: $+12$ and $-8$ .

(a) $(x - 8)(x + 6)$ ; (b) $(x + 12)(x - 8)$

Other Orders and Two Variables

Example 6: Rewrite first

Factorise (a) $-6 + x + x^2$ ; (b) $x^2 + 2xy - 15y^2$ ; (c) $a^2 - ab - 6b^2$ .

Solution

(a) In descending powers: $x^2 + x - 6$ . Difference $1$ : $2, 3$ ; the larger positive: $+3$ , $-2$ .

(b) Product $-15$ , sum $2$ : $+5$ and $-3$ , each with $y$ . Check: $5y \times (-3y) = -15y^2$ and $5xy - 3xy = 2xy$ . ✓

(a) $(x + 3)(x - 2)$ ; (b) $(x + 5y)(x - 3y)$ ; (c) $(a - 3b)(a + 2b)$

Deciding the Signs First

Lessons 3.9 and 3.10 together give four sign patterns. Look at the signs of $c$ and $b$ before you list any pairs:

$c$	$b$	The two numbers	Form
positive	positive	both positive	$(x + p)(x + q)$
positive	negative	both negative	$(x - p)(x - q)$
negative	positive	opposite signs, the larger positive	$(x + \text{larger})(x - \text{smaller})$
negative	negative	opposite signs, the larger negative	$(x - \text{larger})(x + \text{smaller})$

Example 7: The same numbers, four sign patterns

Factorise (a) $x^2 - 9x + 18$ ; (b) $x^2 + 3x - 18$ ; (c) $x^2 - 3x - 18$ ; (d) $x^2 + 9x + 18$ .

Solution

(a) $c > 0$ , $b < 0$ : both negative. Sum $9$ from a pair of $18$ : $3, 6$ . So $-3$ , $-6$ .

(b) $c < 0$ , $b > 0$ : opposite signs, larger positive. Difference $3$ : $3, 6$ . So $+6$ , $-3$ .

(d) $c > 0$ , $b > 0$ : both positive. So $+3$ , $+6$ .

(a) $(x - 3)(x - 6)$ ; (b) $(x + 6)(x - 3)$ ; (c) $(x - 6)(x + 3)$ ; (d) $(x + 3)(x + 6)$

Finding $k$ , and Quadratics That Do Not Factorise

Example 8: All the values of $k$

Find all the integers $k$ for which $x^2 + kx - 12$ can be factorised into two brackets with integers.

Solution

$k$ is the sum of two integers with product $-12$ : $1, -12$ ( $-11$ ); $-1, 12$ ( $11$ ); $2, -6$ ( $-4$ ); $-2, 6$ ( $4$ ); $3, -4$ ( $-1$ ); $-3, 4$ ( $1$ ).

$k = 1, 4, 11, -1, -4$ or $-11$

Example 9: No pair works

Can $x^2 + 3x - 5$ be factorised?

Solution

We need product $-5$ and sum $3$ . The only pairs are $1, -5$ (sum $-4$ ) and $-1, 5$ (sum $4$ ).

No pair has sum $3$ , so $x^2 + 3x - 5$ cannot be factorised into brackets with integers.

Example 10: A rectangle

A rectangle has area $(x^2 + 2x - 15)$ cm², and one side is $(x + 5)$ cm. (a) Find the other side. (b) What must be true of $x$ ?

Solution

(a) Product $-15$ , sum $2$ : $+5$ and $-3$ . So $x^2 + 2x - 15 = (x + 5)(x - 3)\text{,}$ and the other side is $(x - 3)$ cm.

(b) A length must be positive, so $x - 3 > 0$ , that is $x > 3$ .

Check with $x = 5$ : area $25 + 10 - 15 = 20$ cm², and $10 \times 2 = 20$ . ✓

(a) $(x - 3)$ cm; (b) $x > 3$

Common Mistakes

Giving both numbers the same sign: $x^2 + 5x - 14 \ne (x + 7)(x + 2)\text{,}$ whose number term is $+14$ . It is $(x + 7)(x - 2)$ .
Giving the larger number the wrong sign: $x^2 + 2x - 8 = (x + 4)(x - 2)\text{;}$ $(x - 4)(x + 2)$ is $x^2 - 2x - 8$ .
Looking for a pair whose sum is the size of $b$ : when $c$ is negative it is the difference of the sizes.
Not writing the terms in descending powers before starting: $-6 + x + x^2$ is $x^2 + x - 6$ .

Practice

Every question in Questions 1 to 7 has a negative number term. Check each answer by expanding it.

Question 1

Factorise (a) $x^2 + x - 56$ ; (b) $x^2 + 2x - 63$ ; (c) $x^2 + 7x - 18$ .

ShowHide solution

(a) Pairs of $56$ with difference $1$ : $7, 8$ . Larger positive: $+8$ , $-7$ .

(b) Pairs of $63$ with difference $2$ : $7, 9$ . Larger positive: $+9$ , $-7$ .

(a) $(x + 8)(x - 7)$ ; (b) $(x + 9)(x - 7)$ ; (c) $(x + 9)(x - 2)$

Question 2

Factorise (a) $x^2 - 5x - 14$ ; (b) $x^2 - 4x - 21$ ; (c) $x^2 - 2x - 24$ .

ShowHide solution

(a) Pairs of $14$ : $1, 14$ ( $13$ ); $2, 7$ ( $5$ ) ✓. Larger negative: $-7$ , $+2$ .

(b) Pairs of $21$ : $1, 21$ ( $20$ ); $3, 7$ ( $4$ ) ✓. Larger negative: $-7$ , $+3$ .

(a) $(x - 7)(x + 2)$ ; (b) $(x - 7)(x + 3)$ ; (c) $(x - 6)(x + 4)$

Question 3

Factorise (a) $x^2 + x - 30$ ; (b) $x^2 - x - 42$ .

ShowHide solution

(a) Difference $1$ : $5, 6$ . Larger positive: $+6$ , $-5$ .

(b) Difference $1$ : $6, 7$ . Larger negative: $-7$ , $+6$ .

(a) $(x + 6)(x - 5)$ ; (b) $(x - 7)(x + 6)$

Question 4

Factorise (a) $x^2 + 9x - 22$ ; (b) $x^2 - 10x - 11$ ; (c) $x^2 - 5x - 36$ .

ShowHide solution

(a) Pairs of $22$ : $1, 22$ ( $21$ ); $2, 11$ ( $9$ ) ✓. $+11$ , $-2$ .

(b) Pairs of $11$ : only $1, 11$ ( $10$ ) ✓. $-11$ , $+1$ .

(a) $(x + 11)(x - 2)$ ; (b) $(x - 11)(x + 1)$ ; (c) $(x - 9)(x + 4)$

Question 5

Factorise $x^2 + 3x - 28$ using a multiplication frame.

ShowHide solution

Pairs of $28$ : $1, 28$ ; $2, 14$ ; $4, 7$ (difference $3$ ) ✓. Larger positive: $+7$ , $-4$ .

×	$x$	$+7$
$x$	$x^2$	$+7x$
$-4$	$-4x$	$-28$

$x^2 + 7x - 4x - 28 = x^2 + 3x - 28$

$(x - 4)(x + 7)$

Question 6

Factorise (a) $-72 + y + y^2$ ; (b) $p^2 - 3p - 40$ .

ShowHide solution

(a) $y^2 + y - 72$ : difference $1$ , so $8, 9$ ; larger positive: $+9$ , $-8$ .

(b) Pairs of $40$ : $1, 40$ ; $2, 20$ ; $4, 10$ ; $5, 8$ (difference $3$ ) ✓. Larger negative: $-8$ , $+5$ .

(a) $(y + 9)(y - 8)$ ; (b) $(p - 8)(p + 5)$

Question 7

Factorise (a) $x^2 + 4xy - 12y^2$ ; (b) $m^2 - 2mn - 35n^2$ .

ShowHide solution

(a) Product $-12$ , sum $4$ : $+6$ and $-2$ , each with $y$ .

(b) Product $-35$ , sum $-2$ : $-7$ and $+5$ , each with $n$ .

(a) $(x + 6y)(x - 2y)$ ; (b) $(m - 7n)(m + 5n)$

Question 8

Decide the signs first, then factorise. (a) $x^2 - 11x + 24$ ; (b) $x^2 + 5x - 24$ ; (c) $x^2 - 10x - 24$ ; (d) $x^2 + 11x + 24$

ShowHide solution

(a) $c > 0$ , $b < 0$ : both negative, sum $11$ : $3, 8$ .

(b) $c < 0$ , $b > 0$ : opposite, larger positive, difference $5$ : $3, 8$ . So $+8$ , $-3$ .

(d) $c > 0$ , $b > 0$ : both positive, sum $11$ : $3, 8$ .

(a) $(x - 3)(x - 8)$ ; (b) $(x + 8)(x - 3)$ ; (c) $(x - 12)(x + 2)$ ; (d) $(x + 3)(x + 8)$

Question 9

Find all the integers $k$ for which $x^2 + kx - 20$ can be factorised into two brackets with integers.

ShowHide solution

Pairs with product $-20$ and their sums: $1, -20$ ( $-19$ ); $-1, 20$ ( $19$ ); $2, -10$ ( $-8$ ); $-2, 10$ ( $8$ ); $4, -5$ ( $-1$ ); $-4, 5$ ( $1$ ).

$k = 1, 8, 19, -1, -8$ or $-19$

Question 10

Which of these can be factorised? Factorise those that can. (a) $x^2 + 4x - 6$ ; (b) $x^2 + 3x - 10$ ; (c) $x^2 - 2x - 7$

ShowHide solution

(a) Pairs of $6$ : $1, 6$ (difference $5$ ) and $2, 3$ (difference $1$ ). Neither has difference $4$ : it cannot be factorised.

(b) Pairs of $10$ : $2, 5$ has difference $3$ . $(x + 5)(x - 2)$ .

Only (b): $(x + 5)(x - 2)$ .

Question 11

A rectangular carpet has area $(x^2 - x - 20)$ m² and width $(x + 4)$ m. (a) Find its length. (b) What must be true of $x$ ? (c) Check your answer when $x = 9$ .

ShowHide solution

(a) Product $-20$ , sum $-1$ : $-5$ and $+4$ . $x^2 - x - 20 = (x + 4)(x - 5)\text{,}$ so the length is $(x - 5)$ m.

(b) The length must be positive: $x > 5$ .

(a) $(x - 5)$ m; (b) $x > 5$

Question 12

Each line has one mistake. Find it and correct it. (a) $x^2 + 6x - 27 = (x - 9)(x + 3)\text{;}$ (b) $x^2 - x - 6 = (x - 3)(x - 2)\text{;}$ (c) $x^2 + 7x - 30 = (x + 10)(x + 3)$

ShowHide solution

(a) $(x - 9)(x + 3) = x^2 - 6x - 27\text{.}$ For $+6x$ the larger number is positive: $(x + 9)(x - 3)$ .

(b) $(x - 3)(x - 2) = x^2 - 5x + 6\text{.}$ For $-6$ the signs must be opposite: $(x - 3)(x + 2)$ .

(a) $(x + 9)(x - 3)$ ; (b) $(x - 3)(x + 2)$ ; (c) $(x + 10)(x - 3)$

Key Points

When $c$ is negative, the two numbers have opposite signs.
Find a factor pair of $c$ whose difference is the size of $b$ ; the larger number takes the sign of $b$ .
Decide the signs from $c$ and $b$ before listing pairs.
If no pair works, the quadratic does not factorise with integers.
Check by expanding.

Factorising x² + bx + c When c Is Negative

Before you start

Why the Two Numbers Have Opposite Signs

Factorising x2+bx+cx^2 + bx + cx2+bx+c when ccc is negative

When bbb Is Positive

Example 1: Every pair, with its signs

Example 2: Using the difference

When bbb Is Negative

Example 3: The larger number negative

Example 4: When bbb is 111 or −1-1−1

Example 5: Many pairs

Other Orders and Two Variables

Example 6: Rewrite first

Deciding the Signs First

Example 7: The same numbers, four sign patterns

Finding kkk, and Quadratics That Do Not Factorise

Example 8: All the values of kkk

Example 9: No pair works

Example 10: A rectangle

Common Mistakes

Practice

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Key Points

Factorising $x^2 + bx + c$ when $c$ is negative

When $b$ Is Positive

When $b$ Is Negative

Example 4: When $b$ is $1$ or $-1$

Finding $k$ , and Quadratics That Do Not Factorise

Example 8: All the values of $k$