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How to Solve a Quadratic Equation with the Quadratic Formula
Any equation of the form ax² + bx + c = 0 can be solved with one formula. Learn the quadratic formula, what the discriminant tells you about the roots, and how to find the vertex.
Updated 2 November 2026 · 3 min read
A quadratic equation is any equation you can write as ax² + bx + c = 0, where a isn't zero. The highest power of the unknown is 2, which is what gives its graph the familiar U-shaped parabola. Solving it means finding the values of x — the roots — that make the equation true. There's one formula that cracks every quadratic, no matter how awkward the numbers.
The quadratic formula
Once you've identified the three coefficients — a, b and c — the roots come straight from a single expression:
Take x² − 3x + 2 = 0, so a = 1, b = −3 and c = 2. The part under the root is (−3)² − 4·1·2 = 9 − 8 = 1, and √1 = 1. That gives x = (3 + 1) ÷ 2 = 2 and x = (3 − 1) ÷ 2 = 1. The two roots are x = 2 and x = 1 — the points where the parabola crosses the x-axis.
What the discriminant tells you
The expression under the square root, b² − 4ac, is called the discriminant, and it reveals the nature of the roots before you finish the sum:
- Positive — two different real roots (the parabola crosses the x-axis twice).
- Zero — one repeated real root (the parabola just touches the x-axis).
- Negative — no real roots; the solutions are a pair of complex numbers (the parabola never touches the x-axis).
That square-root step is where quadratics lean on square roots — if the discriminant isn't a perfect square, the answer stays as a surd or becomes a decimal. The quadratic equation solver handles all three cases, returns complex roots when the discriminant is negative, and shows the discriminant and vertex alongside the answer.
Finding the vertex
The vertex is the turning point of the parabola — its lowest point when a is positive, its highest when a is negative. Its x-coordinate is −b ÷ 2a, and you get the y-coordinate by substituting that x back into the equation. For x² − 3x + 2, the vertex sits at x = 3 ÷ 2 = 1.5, exactly halfway between the two roots. That symmetry is always true: the vertex lies on the line of symmetry midway between the roots.
A quick checklist
- Rearrange the equation into the form ax² + bx + c = 0 so one side is zero.
- Read off a, b and c, keeping the signs.
- Work out the discriminant b² − 4ac to see how many real roots to expect.
- Plug everything into the quadratic formula and compute both the + and − versions.
Quadratics are a cornerstone of algebra, and the skills carry over to the rest of the toolkit. If the powers in the formula feel shaky, our guide to exponents is a useful refresher, and the area of a circle walkthrough is another good example of a formula that hinges on squaring a value.
Frequently asked questions
- Why does a quadratic have two solutions?
- The ± in the quadratic formula means you add and subtract the square root, giving two values. Graphically, a parabola usually crosses the x-axis in two places. When the discriminant is zero the two solutions coincide into one repeated root, and when it's negative there are no real crossings at all.
- What if a is zero?
- Then it isn't a quadratic — the x² term vanishes and you're left with the linear equation bx + c = 0, which has a single solution, x = −c ÷ b.
- What are complex roots?
- When the discriminant is negative you'd need the square root of a negative number, which has no real value. The roots are then complex, written as a ± bi, where i is the square root of −1. They come in pairs and still satisfy the equation.