Which of the following is equivalent to (2x + 1)(x - 3)? A) 2x^2

Step by step

The question

Question: simplify (2x + 1)(x − 3).

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

(2x + 1)(x − 3)

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 2x and 1.

Each one takes the whole second bracket.

= 2x(x − 3) + 1(x − 3)

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.

= 2x(x) + 2x(−3) + 1(x) + 1(−3)

− × + = −

x times x is x²: x squared.

x × x = x²

Watch the signs.

2x times x is 2x².

2x times (−3) is −6x.

1 times x is x.

1 times (−3) is −3.

= 2x² − 6x + x − 3

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

x alone means 1x.

The numbers in front: −6 plus 1 is −5.

So −6x + x is −5x.

x = 1x

−6x + x = −5x

So the expression is 2x² − 5x − 3.

= 2x² − 5x − 3

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 × 2 + 1)(2 − 3) = (5)(−1) = −5

Now into our answer.

Answer: 2 × 2² − 5 × 2 − 3 = 8 − 10 − 3

= −5

Both give −5.

Same value, so 2x² − 5x − 3 is right!

Both give −5

5 Pick the choice

Our answer is 2x² − 5x − 3.

That is choice B.

B) 2x² − 5x − 3

6 Why not the others

A is 2x² − 3.

Wrong: forgot the middle parts: −6x and +x.

A) 2x² − 3 ✗ forgot the middle parts: −6x and +x

C is 2x² − 7x − 3.

Wrong: −6x + x is −5x, not −7x.

C) 2x² − 7x − 3 ✗ −6x + x is −5x, not −7x

D is 2x² − 5x + 3.

Wrong: 1 × (−3) is −3, not +3.

D) 2x² − 5x + 3 ✗ 1 × (−3) is −3, not +3

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