Three Ways to Solve a Quadratic Equation

A quadratic equation always has two solutions. Which method you reach for depends on the shape of the equation in front of you — and one of the three works every single time.

Concept What makes an equation quadratic

In a quadratic the highest power of the variable is 2. That is why it always carries two solutions, even when the two happen to be equal.

Example Method 1 — take the square root

Use this when the equation has the form , with no term. Solve .

Take the square root of both sides.

Both signs work:

and

⟹ x = 2 or x = −2

The is essential. Writing only loses half the answer.

Example Method 2 — take out a common factor

Use this when every term shares a factor. Solve .

Both terms contain

, so factor it out:

If a product is zero, at least one factor must be zero:

⟹ x = 0 or x = 3

Never divide both sides by here — that would silently discard the solution .

Theorem Method 3 — the general formula

For the standard form :

Here is the coefficient of , the coefficient of , and the constant. This method solves every quadratic, including those the first two cannot touch.

Example Reading off a, b and c

Arrange each equation from the highest power down, then identify the coefficients.

rearranges to

Watch the signs — a negative coefficient must be carried into the formula as a negative.

Example Applying the formula

Solve , where .

, so:

⟹ x = −1 + √2 or x = −1 − √2

Summary

A quadratic has highest power 2 and always two solutions.
Square root method: for x² = k, giving x = ±√k.
Factoring: take out the common factor, then set each factor to zero.
General formula: works for every quadratic once written as ax² + bx + c = 0.
Arrange the equation first, mind the signs, and check both answers.