How to Solve Quadratic Equations: 4 Methods Explained

Factoring, the square root method, completing the square and the quadratic formula — when to use each, with worked examples.

By AI Math Solver Team · Sep 1, 2026 · 8 min read

A quadratic equation has the form $a x^{2} + b x + c = 0$ with $a \neq = 0$ . There are four standard ways to solve one, and choosing the right method saves a lot of time.

1. Factoring

If the quadratic factors nicely, set each factor equal to zero.

$x^{2} - 5 x + 6 = 0; \Rightarrow; (x - 2) (x - 3) = 0; \Rightarrow; x = 2 or x = 3$

Use it when you can quickly spot two numbers that multiply to $c$ and add to $b$ .

2. The square root method

When there is no $x$ term, isolate $x^{2}$ and take square roots — remembering both signs.

$4 x^{2} = 36; \Rightarrow; x^{2} = 9; \Rightarrow; x = \pm 3$

3. Completing the square

Rewrite the equation as a perfect square. This method explains where the quadratic formula comes from and gives vertex form.

$x^{2} + 6 x + 2 = 0; \Rightarrow; (x + 3)^{2} = 7; \Rightarrow; x = - 3 \pm 7$

4. The quadratic formula

It always works:

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

The discriminant $Δ = b^{2} - 4 a c$ tells you what to expect before you calculate:

Discriminant	Roots
$Δ > 0$	two distinct real roots
$Δ = 0$	one repeated real root
$Δ < 0$	two complex roots

Which method should you pick?

No $x$ term? Use square roots.
Small integer coefficients and $Δ$ a perfect square? Factor.
Otherwise, use the quadratic formula.

Common mistakes

Forgetting the $\pm$ when taking square roots.
Not moving every term to one side before factoring.
Sign errors with $- b$ when $b$ is negative.

Want to check your work? Try our Quadratic Formula Calculator — every answer is verified by substitution.

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