# Quadratic Formula

How to solve quadratic equations by using the quadratic formula: formula, proof, examples, and their solutions.

## Formula

For any quadratic equation (*a* ≠ 0),*x* can be found by using the quadratic formula.

## Proof

Prove the quadratic formula

by completing the square.

Move +*c* to the right side.

Divide both sides by *a*.

Change +(*b*/*a*)*x* to +2⋅*x*⋅[*b*/(2*a*)].

Write +[*b*/(2*a*)]^{2} on both sides.

Factor the left side by using the formula:*x*^{2} + 2⋅*x*⋅[*b*/(2*a*)] + [*b*/(2*a*)]^{2} = (*x* + [*b*/(2*a*)])^{2}.

Change -(*c*/*a*) to -[(*c*⋅4*a*)/(*a*⋅4*a*)].

Then add it with +*b*^{2}/4*a*^{2}.

Square root both sides.

Move +*b*/(2*a*) to the right side.

Then *x* = [(-*b* ± √*b*^{2} - 4*ac*)]/(2*a*).

## Example 1

Put *a* = 1, *b* = 3, *c* = -2 into the formula.

## Example 2

Put *a* = 4, *b* = -1, *c* = +5 into the formula.

√-79 is not a real number

because the number inside the radical sign

cannot be (-).

So this equation has no real roots.

If you know about complex numbers,

there's a way to find the roots.

Complex roots of a quadratic equation