Which expression is equivalent to (x + 4)(x - 4)? A) x^2 + 16 {4 ×… Step by Step

Step by step

The question

Question: simplify (x + 4)(x − 4).

Simplify means: write it shorter, with the same value.

(x + 4)(x − 4)

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Split the first bracket

Each part of the first bracket multiplies the whole second bracket.

(a + b)(c + d) = a(c + d) + b(c + d)

The parts of the first bracket are x and 4.

Each one takes the whole second bracket.

= x(x − 4) + 4(x − 4)

2 Open the brackets

What is in front of a bracket multiplies every part inside.

a(b + c) = ab + ac

So we write each part times what is in front.

= x(x) + x(−4) + 4(x) + 4(−4)

− × + = −

x times x is x²: x squared.

x × x = x²

Watch the signs.

x times x is x².

x times (−4) is −4x.

4 times x is 4x.

4 times (−4) is −16.

= x² − 4x + 4x − 16

3 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

Each colour is one kind.

like terms = same letters,

same powers

−4x + 4x is 0.

−4x + 4x = 0

So the expression is x² − 16.

= x² − 16

4 Check

Let's check.

We pick a small number for each letter: x = 2.

Any number works, and a small one keeps the sums easy.

We put it into the question.

Question: (2 + 4)(2 − 4) = (6)(−2) = −12

Now into our answer.

Answer: 2² − 16 = 4 − 16 = −12

Both give −12.

Same value, so x² − 16 is right!

Both give −12

5 Pick the choice

Our answer is x² − 16.

That is choice C.

C) x² − 16

6 Why not the others

A is x² + 16.

Wrong: 4 × (−4) is −16, not +16.

A) x² + 16 ✗ 4 × (−4) is −16, not +16

B is x² − 8.

Wrong: 4 × 4 is 16, not 8.

B) x² − 8 ✗ 4 × 4 is 16, not 8

D is x² − 8x − 16.

Wrong: −4x + 4x is 0, not −8x.

D) x² − 8x − 16 ✗ −4x + 4x is 0, not −8x

Let's look back.

First: split the first bracket.

Then: open the brackets.

Then: group like terms.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer