Quadratics with complex solutions, step by step

When the discriminant is negative the parabola misses the axis and the roots leave the real numbers.

The question

Solve x² − 2x + 50 = 0.

Solve x² − 2x + 50 = 0

b² − 4ac = −196

x = (2 ± √−196) ÷ 2

x = (2 ± 14i) ÷ 2

x = 1 ± 7i

Work out the discriminant first: b² − 4ac = (−2)² − 4(50) = 4 − 200 = −196. It is negative, so there are no real roots.
Put it in the formula: x = (2 ± √−196) ÷ 2.
√−196 = 14i, because √(−1) is i. So x = (2 ± 14i) ÷ 2.
Divide both parts by 2: x = 1 ± 7i.

How it works.

01
Work out the discriminant first: b² − 4ac = (−2)² − 4(50) = 4 − 200 = −196. It is negative, so there are no real roots.
02
Put it in the formula: x = (2 ± √−196) ÷ 2.
03
√−196 = 14i, because √(−1) is i. So x = (2 ± 14i) ÷ 2.
04
Divide both parts by 2: x = 1 ± 7i.