Which of the following is equivalent to (x - 5)^2? A) x^2

Step by step

The question

Question: simplify (x − 5)².

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

(x − 5)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 − 5)² is (x − 5) times (x − 5).

( )² = ( )( )

Careful!

We can't just square each part.

x² + 25 is wrong.

The middle part is missing.

Careful: not (x − 5)² = x² + 25 ✗

So we write the bracket two times.

= (x − 5)(x − 5)

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 −5.

Each one takes the whole second bracket.

= x(x − 5) − 5(x − 5)

3 Open the brackets

The minus sign in front of the 5 goes with it.

So we multiply by −5.

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(−5) + (−5)(x) + (−5)(−5)

− × − = +

x times x is x²: x squared.

x × x = x²

Watch the signs.

x times x is x².

x times (−5) is −5x.

(−5) times x is −5x.

(−5) times (−5) is 25.

= x² − 5x − 5x + 25

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

The numbers in front: −5 minus 5 is −10.

So −5x − 5x is −10x.

−5x − 5x = −10x

So the expression is x² − 10x + 25.

= x² − 10x + 25

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 − 5)² = (−3)² = 9

Now into our answer.

Answer: 2² − 10 × 2 + 25 = 4 − 20 + 25 = 9

Both give 9.

Same value, so x² − 10x + 25 is right!

Both give 9

6 Pick the choice

Our answer is x² − 10x + 25.

That is choice B.

B) x² − 10x + 25

7 Why not the others

A is x² − 25.

Wrong: squared each part: lost −5x − 5x.

A) x² − 25 ✗ squared each part: lost −5x − 5x

C is x² − 10x − 25.

Wrong: (−5) × (−5) is +25, not −25.

C) x² − 10x − 25 ✗ (−5) × (−5) is +25, not −25

D is x² − 5x + 25.

Wrong: kept one −5x: it is −5x − 5x = −10x.

D) x² − 5x + 25 ✗ kept one −5x: it is −5x − 5x = −10x

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