What are the solutions of 2x^2 + 5x

Step by step

The question

Question: solve 2x² + 5x − 3 = 0.

2x² + 5x − 3 = 0

There is an x².

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Find two numbers

There is a number in front of x², but it can still factor.

The rule: we split the middle into two parts.

Their numbers must multiply to first × last.

Here the first number is 2 and the last is −3.

2 × −3 = −6.

So we need two numbers that multiply to −6.

They must also add to the middle number, 5.

2 × (−3) = −6

? × ? = −6

? + ? = 5

They multiply to a minus number, so one number is plus and one is minus.

They add to a plus number, so the bigger one is plus.

When we add a plus and a minus, the bigger one wins.

× is − → one +, one −; bigger one +

We list the pairs that make 6, smallest first.

For each pair, we check two things: they multiply to 6, and the bigger take away the smaller is 5.

One is minus, so adding them is a take away.

multiply to 6 bigger − smaller = 5

1 × 6 = 6.

6 − 1 = 5.

Both work!

1 × 6 = 6 ✓ 6 − 1 = 5 ✓

So the two numbers are 1 and 6.

With the signs: −1 and 6.

1 and 6

2 Split the middle

We write 5x as −x + 6x.

It is the same amount.

2x² − x + 6x − 3 = 0

3 Group in pairs

We put the first two together, and the last two together.

(2x² − x) + (6x − 3) = 0

2x² and −x both have x in them.

So from the first pair we take out x.

6x and −3 both have 3 in them.

So from the second pair we take out 3.

Both leave the same bracket.

(2x² − x) = x(2x − 1)

(6x − 3) = 3(2x − 1)

Now both parts have 2x − 1 in them.

x(2x − 1) + 3(2x − 1) = 0

4 Same bracket out

2x − 1 is in both parts, so we take it out once.

What is left, x + 3, goes in the other bracket.

ab + cb = b(a + c)

So we get (2x − 1)(x + 3).

(2x − 1)(x + 3) = 0

5 One bracket is 0

Two numbers multiply to 0.

That only happens if one of them is 0.

So one bracket must be 0.

A × B = 0 → A = 0 or B = 0

We make each bracket 0.

Two small equations.

2x − 1 = 0 or x + 3 = 0

In 2x − 1 = 0, we add 1 on both sides.

In x + 3 = 0, we take away 3 on both sides.

2x − 1 + 1 = 0 + 1 or x + 3 − 3 = 0 − 3

Now we have 2x = 1 or x = −3.

2x = 1 or x = −3

To remove times 2, we divide both sides by 2.

So x = 1/2 or x = −3.

x = 12 or x = −3

6 Check

Let's check: put x = 1/2 into the left side.

x = 12: 2(12)² + 5(12) − 3 = 12 + 52 − 3 = 0

We get 0, the same as the right side.

It works!

Now put x = −3 into the left side.

x = −3: 2(−3)² + 5(−3) − 3 = 18 − 15 − 3

= 0

We get 0 again, the same as the right side.

So both answers are right!

7 Pick the choice

Our answer is x = 1/2 or x = −3.

That is choice A.

A) −3 and 12

8 Why not the others

B is −3 and 1.

Wrong: 2x − 1 = 0 means 2x = 1: we still had to divide by 2.

B) −3 and 1 ✗ 2x − 1 = 0 means 2x = 1: we still had to divide by 2

C is −3/2 and 1.

Wrong: (2x + 3)(x − 1) gives + x, not + 5x.

C) −32 and 1 ✗ (2x + 3)(x − 1) gives + x, not + 5x

D is −1/2 and 3.

Wrong: signs swapped.

D) −12 and 3 ✗ signs swapped

Let's look back.

First: find two numbers.

Then: split the middle.

Then: group in pairs.

Then: same bracket out.

Then: one bracket is 0.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer