Choose a method: x^2 + x - 12 = 0 Step by Step

Step by step

The question

Question: choose a method and solve x² + x − 12 = 0.

x² + x − 12 = 0

1 Choose the method

We ask in order.

First: is there no x term?

No, there is one.

1. no x term, or ( )² = k? ✗

x² has no number in front, and two whole numbers work.

So it factors.

So: factoring.

2. factors easily? ✓ factoring

2 Factor

We need two numbers.

They multiply to −12, and they add to 1.

? × ? = −12

? + ? = 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 12, smallest first.

We take the smaller from the bigger.

We need 1.

3 and 4 give 1.

Yes!

With the signs: −3 and 4.

1 × 12: 12 − 1 = 11

2 × 6: 6 − 2 = 4

3 × 4: 4 − 3 = 1 ✓

So it is (x − 3)(x + 4) = 0.

(x − 3)(x + 4) = 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 − 3 = 0 or x + 4 = 0

In x − 3 = 0, we add 3 on both sides.

In x + 4 = 0, we take away 4 on both sides.

x − 3 + 3 = 0 + 3 or x + 4 − 4 = 0 − 4

So x = 3 or x = −4.

x = 3 or x = −4

4 Check

Let's check: put x = 3 into the left side.

x = 3: 3² + 3 − 12 = 9 + 3 − 12 = 0

We get 0, the same as the right side.

It works!

Now put x = −4 into the left side.

x = −4: (−4)² + (−4) − 12 = 16 − 4 − 12 = 0

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

So both answers are right!

Let's look back.

First: choose the method.

Then: factor.

Then: one bracket is 0.

Last: check.

Answer