How to solve quadratic equations, step by step

A quadratic equation is any equation you can write in the form $ax^2 + bx + c = 0$, where $a \ne 0$. The goal is always the same: find the values of $x$ that make it true. Those are the equation's roots.

Method 1: Factoring

If the quadratic factors neatly, this is the fastest route. You rewrite it as a product of two brackets, then use the fact that if a product is zero, one of its factors must be zero.

Method 2: The quadratic formula

When factoring isn't obvious, the quadratic formula always works. For $ax^2 + bx + c = 0$:

The part under the root, $b^2 - 4ac$, is the discriminant. If it's positive there are two real roots; zero gives one; negative means the roots are complex.

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Method 3: Completing the square

This method rewrites the equation as a perfect square plus a constant. It's the most work by hand, but it's how the quadratic formula is derived, and it's essential for tasks like finding a parabola's vertex.

Take $x^2 + 6x + 5 = 0$. Move the constant, halve the coefficient of $x$ and square it, then rewrite: $(x+3)^2 = 4$, which gives $x = -1$ or $x = -5$.

Which method should you use?

• Factoring: try first; fastest when the numbers are friendly.
• Quadratic formula: the dependable fallback that always works.
• Completing the square: when you also need the vertex or a proof.

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