Magpie Tuition · Higher Maths

Distributing a negative term

Allow 7–10 minutes: about 3–4 minutes reading, then practice. No calculator needed.

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Overview

Why this catches people

The minus sign outside a bracket belongs to the whole bracket. Removing the bracket means multiplying every term inside it. It is easy to change the first term and leave the second untouched.

Key idea

The rule

Write subtraction inside a bracket as a signed term when it helps:

-k(a-b)=(-k)\times a+(-k)\times(-b)=-ka+kb.

Multiply the outside number by each term. A negative times a positive is negative; a negative times a negative is positive. Multiply coefficients—do not add them.

A minus sign on its own means multiplication by $-1$ :

-(a+b)=-a-b,\qquad -(a-b)=-a+b.

Worked example

Worked examples

1. Subtracting a bracket

-(x-9)=(-1)\times x+(-1)\times(-9)=-x+9.

Both terms change sign. The result is not $-x-9$ .

2. Both terms inside are positive

-4(3x+2)=(-4)\times3x+(-4)\times2=-12x-8.

The coefficient of $x$ is $(-4)\times3=-12$ , not $-4+3=-1$ .

3. One term inside is negative

-5(2x-7)=(-5)\times2x+(-5)\times(-7)=-10x+35.

For a quick check, put $x=0$ : the original gives $-5(-7)=35$ ; the answer also gives $35$ . A check at one value can catch a slip, but does not prove two expressions are identical.

Common mistake

Common trap

$-(a-b)$ means subtract all of $a-b$ . Changing only the first sign gives $-a-b$ , which is a different expression. If your answer differs from the solution, write the two separate multiplications to locate the step that needs attention.

Practice

Your Turn

Practice only — these are not your fresh-check questions. Show each multiplication before simplifying.

Expand $-(x+8)$ .
Expand $-(4x-9)$ .
Expand $-3(5x+2)$ .
Expand $-6(2x-5)$ .
Expand $-2(-4x+7)$ .
A student writes $-4(2x-3)=-8x-3$ . Identify the missed operation and correct the working.

Answers

Answers with working

Show the answers

$(-1)\times x+(-1)\times8=\boxed{-x-8}$ .
$(-1)\times4x+(-1)\times(-9)=\boxed{-4x+9}$ .
$(-3)\times5x+(-3)\times2=\boxed{-15x-6}$ .
$(-6)\times2x+(-6)\times(-5)=\boxed{-12x+30}$ .
$(-2)\times(-4x)+(-2)\times7=\boxed{8x-14}$ .
The $-3$ also needs multiplying by $-4$ . The full line is $(-4)\times2x+(-4)\times(-3)=\boxed{-8x+12}$ . The original answer fits a possible first-term-only slip; it does not tell us why the student made it.

Before you move on

Ready to move on?

Can you explain why every term is multiplied, why two negatives give the positive term here, and why coefficients are multiplied rather than added? If not, revisit the matching example. Completing this practice is preparation for a fresh check, not proof of mastery.

Magpie Tuition original resource, reviewed for V1. It has not yet been tried out with students, so if something is unclear, tell your tutor. Printing is fine; please don't republish it.