Quadratic Formula

Quadratic Formula

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

Formula

If ax^2 + bx + c = 0 (a ≠ 0), then x = (-b ± sqrt(b^2 - 4ac))/2a.

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

Proof

Given: ax^2 + bx + c = 0 (a ≠ 0) Prove: x = (-b ± sqrt(b^2 - 4ac))/2a

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/(2a)].

Write +[b/(2a)]2 on both sides.

Factor the left side by using the formula:
x2 + 2⋅x⋅[b/(2a)] + [b/(2a)]2 = (x + [b/(2a)])2.

Change -(c/a) to -[(c⋅4a)/(a⋅4a)].

Then add it with +b2/4a2.

Square root both sides.

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

Then x = [(-b ± √b2 - 4ac)]/(2a).

Example 1

Solve the given equation. x^2 + 3x - 2 = 0

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

Example 2

Solve the given equation. 4x^2 - x + 5 = 0

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