Which expression is equivalent to (x + 3)^2? A) x^2 + 9 {squared… Step by Step

Step by step

The question

Question: simplify (x + 3)².

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

(x + 3)²

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Write the square out

A square means the bracket times itself.

(x + 3)² is (x + 3) times (x + 3).

( )² = ( )( )

Careful!

We can't just square each part.

x² + 9 is wrong.

The middle part is missing.

Careful: not (x + 3)² = x² + 9 ✗

So we write the bracket two times.

= (x + 3)(x + 3)

2 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 3.

Each one takes the whole second bracket.

= x(x + 3) + 3(x + 3)

3 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(3) + 3(x) + 3(3)

x times x is x²: x squared.

x × x = x²

x times x is x².

x times 3 is 3x.

3 times x is 3x.

3 times 3 is 9.

= x² + 3x + 3x + 9

4 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

3x + 3x is 6x.

It is like 3 apples plus 3 apples: that is 6 apples.

3x + 3x = 6x

So the expression is x² + 6x + 9.

= x² + 6x + 9

5 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 + 3)² = (5)² = 25

Now into our answer.

Answer: 2² + 6 × 2 + 9 = 4 + 12 + 9 = 25

Both give 25.

Same value, so x² + 6x + 9 is right!

Both give 25

6 Pick the choice

Our answer is x² + 6x + 9.

That is choice B.

B) x² + 6x + 9

7 Why not the others

A is x² + 9.

Wrong: squared each part: lost 3x + 3x.

That is the Careful mistake from before.

A) x² + 9 ✗ squared each part: lost 3x + 3x

C is x² + 3x + 9.

Wrong: kept one 3x: it is 3x + 3x = 6x.

C) x² + 3x + 9 ✗ kept one 3x: it is 3x + 3x = 6x

D is 2x + 6.

Wrong: 2(x + 3): doubled instead of squaring.

D) 2x + 6 ✗ 2(x + 3): doubled instead of squaring

Let's look back.

First: write the square out.

Then: split the first bracket.

Then: open the brackets.

Then: group like terms.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer