Simplify x^2 - 2x + 5 = 0 using i

Step by step

The question

Question: solve x² − 2x + 5 = 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 = −2, c = 5.

We put the numbers in, each in brackets.

−2 squared is 4.

4 times 5 is 20.

4 take away 20 is −16.

Under the root we have −16.

It is negative, so we use i.

√(−16) is √16 times √(−1): 4i.

Both parts on top divide by 2.

2 divided by 2 is 1.

4i divided by 2 is 2i.

So x = 1 + 2i or x = 1 − 2i.

2 Check

First, (1 + 2i)² is (1 + 2i)(1 + 2i).

Each part times each part.

4i² is minus 4.

So it is −3 + 4i.

Let's check 1 + 2i.

(1 + 2i)² is −3 + 4i: each part times each part, and i² is minus 1.

We get 0.

Now 1 − 2i.

We get 0 again.

Both answers work!

Let's look back.

First: use the formula.

Last: check.