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.
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.
Quadratic Equation: Complex Roots