Quadratic Formula

How to solve a quadratic equation by using the quadratic formula: formula, 2 examples, and their solutions.

Formula

Formula

For a quadratic equation
ax2 + bx + c = 0
(a ≠ 0),

x = [-b ± √b2 - 4ac] / 2a.

This is the quadratic formula.

Quadratic Formula: Proof

Example 1

Example

Solution

The given quadratic equation is
1x2 + 3x - 2 = 0.

a = 1
b = 3
c = -2

Then, by the quadratic formula,
x = [-3 ± √32 - 4⋅1⋅(-2)] / [2⋅1].

32 = 9
-4⋅1⋅(-2) = +8

2⋅1 = 2

9 + 8 = 17

So x = [-3 ± √17]/2.

Example 2

Example

Solution

The given quadratic equation is
4x2 - x + 5 = 0.

a = 4
b = -1
c = +5

Then, by the quadratic formula,
x = [+1 ± √(-1)2 - 4⋅4⋅5] / [2⋅4].

+1 = 1

(-1)2 = 1
-4⋅4⋅5 = -80

2⋅4 = 8

1 - 80 = -79

See √-79.

The number in the radical sign (radicand)
is minus.

But if the radicand is minus,
it's not a real number.

So the quadratic equation has no real roots.

So [No real roots] is the answer.

To see how to write x in complex number,
click the link below.

Complex roots of a quadratic equation