Choose a method: x^2 = 4x + 21 Step by Step

Step by step

The question

Question: choose a method and solve x² = 4x + 21.

x² = 4x + 21

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 Get 0 on one side

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

So we take away 4x + 21 on both sides.

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

x² − 4x − 21 = 4x + 21 − 4x − 21

Now we have x² − 4x − 21 = 0.

x² − 4x − 21 = 0

3 Factor

We need two numbers.

They multiply to −21, and they add to −4.

? × ? = −21

? + ? = −4

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

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

× is − → one +, one −; bigger one −

We list the pairs that make 21, smallest first.

We take the smaller from the bigger.

We need 4.

3 and 7 give 4.

Yes!

With the signs: 3 and −7.

1 × 21: 21 − 1 = 20

3 × 7: 7 − 3 = 4 ✓

So it is (x + 3)(x − 7) = 0.

(x + 3)(x − 7) = 0

4 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 − 7 = 0

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

In x − 7 = 0, we add 7 on both sides.

x + 3 − 3 = 0 − 3 or x − 7 + 7 = 0 + 7

So x = −3 or x = 7.

x = −3 or x = 7

5 Check

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

x = −3, left: (−3)² = 9

And the right side.

right: 4(−3) + 21 = −12 + 21 = 9

Both sides are 9.

It works!

Now put x = 7 into the left side.

x = 7, left: 7² = 49

And the right side.

right: 4(7) + 21 = 28 + 21 = 49

Both sides are 49.

So both answers are right!

Let's look back.

First: choose the method.

Then: get 0 on one side.

Then: factor.

Then: one bracket is 0.

Last: check.

Answer