Solve x^2 + 3x = 2x + 6 Step by Step

Step by step

The question

Question: solve x² + 3x = 2x + 6.

x² + 3x = 2x + 6

1 Get 0 on one side

To solve it, we want 0 on the right side.

So we take away 2x + 6 on both sides.

We do it on both sides, so they stay equal.

x² + 3x − 2x − 6 = 2x + 6 − 2x − 6

Now we have x² + x − 6 = 0.

x² + x − 6 = 0

2 Factor

We need two numbers.

They multiply to −6, and they add to 1.

? × ? = −6

? + ? = 1

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.

× is − → one +, one −; bigger one +

We list the pairs that make 6, smallest first.

We take the smaller from the bigger.

We need 1.

2 and 3 give 1.

Yes!

With the signs: −2 and 3.

1 × 6: 6 − 1 = 5

2 × 3: 3 − 2 = 1 ✓

So it is (x − 2)(x + 3) = 0.

(x − 2)(x + 3) = 0

3 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.

x − 2 = 0 or x + 3 = 0

In x − 2 = 0, we add 2 on both sides.

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

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

So x = 2 or x = −3.

x = 2 or x = −3

4 Check

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

x = 2, left: 2² + 3(2) = 4 + 6 = 10

And the right side.

right: 2(2) + 6 = 4 + 6 = 10

Both sides are 10.

It works!

Now put x = −3 into the left side.

x = −3, left: (−3)² + 3(−3) = 9 − 9 = 0

And the right side.

right: 2(−3) + 6 = −6 + 6 = 0

Both sides are 0.

So both answers are right!

Let's look back.

First: get 0 on one side.

Then: factor.

Then: one bracket is 0.

Last: check.

Answer