Simplify x^2 + 4x + 13 = 0 using i

Step by step

The question

Question: solve x² + 4x + 13 = 0 using i.

No real number times itself is negative.

So we use a new number: i, the square root of minus 1.

So i squared is minus 1.

1 Use the formula

It does not split into brackets, so we use the formula.

Here a = 1, b = 4, c = 13.

We put the numbers in, each in brackets.

4 squared is 16.

4 times 13 is 52.

16 take away 52 is −36.

Under the root we have −36.

It is negative, so we use i.

√(−36) is √36 times √(−1): 6i.

Both parts on top divide by 2.

−4 divided by 2 is −2.

6i divided by 2 is 3i.

So x = −2 + 3i or x = −2 − 3i.

2 Check

First, (−2 + 3i)² is (−2 + 3i)(−2 + 3i).

Each part times each part.

9i² is minus 9.

So it is −5 − 12i.

Let's check −2 + 3i.

(−2 + 3i)² is −5 − 12i: each part times each part, and i² is minus 1.

We get 0.

Now −2 − 3i.

We get 0 again.

Both answers work!

Let's look back.

First: use the formula.

Last: check.