A quadratic equation is any equation that can be written as ax²+bx+c=0, where a, b and c are numbers and a is not zero. Solving it means finding the value or values of x that make the equation true. There are a few ways to do this — factoring, completing the square, graphing — but the quadratic formula is the one method that works on every quadratic equation without exception, so it's worth understanding properly even if you also learn the shortcuts.
The quadratic formula
For any equation ax²+bx+c=0, the two solutions are given by:
x = (−b ± √(b²−4ac)) / 2a
The ± symbol means you calculate the expression twice — once adding the square root, once subtracting it — which is why a quadratic equation has up to two solutions. You plug in the values of a, b and c directly from your equation, in the order they appear in ax²+bx+c=0.
The discriminant: what it tells you before you even solve
The expression under the square root, b²−4ac, is called the discriminant. Calculating it first tells you what kind of answer to expect, which is useful for checking your work:
- Discriminant > 0: two distinct real solutions — the most common case
- Discriminant = 0: exactly one real solution (a repeated root) — the equation touches zero at a single point
- Discriminant < 0: no real solutions — only complex solutions, since you'd be taking the square root of a negative number
If your discriminant comes out negative and you were expecting a real-world answer (like a dimension or a time), that's usually a sign to double-check the original equation rather than push forward into complex numbers.
Worked example
Solve 2x²+5x−3=0.
Here a=2, b=5, c=−3. First, the discriminant: b²−4ac = 5² − 4(2)(−3) = 25 + 24 = 49. Since 49 is positive, expect two distinct real solutions, and since it's a perfect square (49 = 7²), expect the roots to be clean rational numbers rather than messy decimals.
Now the formula: x = (−5 ± √49) / (2×2) = (−5 ± 7) / 4.
Taking the plus case: x = (−5+7)/4 = 2/4 = 0.5.
Taking the minus case: x = (−5−7)/4 = −12/4 = −3.
So the two solutions are x = 0.5 and x = −3. You can verify either one by substituting back into the original equation: 2(0.5)²+5(0.5)−3 = 0.5+2.5−3 = 0. ✓
Common mistakes
The most frequent error is forgetting that b is squared before the 4ac is subtracted — b²−4ac, not (b−4ac)² or b²−4a+c. The second most common mistake is sign errors when a or c is negative: if c=−3, then −4ac becomes −4a(−3) = +12a, and it's easy to drop that sign flip under time pressure. Writing out each substitution on its own line, rather than doing it in your head, catches both of these.
When factoring is faster
If a quadratic factors into whole numbers, factoring is quicker than the formula. For x²−5x+6=0, you're looking for two numbers that multiply to 6 and add to −5: those are −2 and −3, giving (x−2)(x−3)=0, so x=2 or x=3. The catch is that most quadratics you'll meet outside a textbook — ones with real-world coefficients like 2.3x²−7.1x+1.4=0 — simply don't factor into clean numbers, which is exactly when the quadratic formula earns its place as the reliable fallback.
Skip the manual arithmetic entirely with the Quadratic Equation Solver, which shows the discriminant and both roots (real or complex) instantly for any a, b and c you enter.
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Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b²−4ac)) / 2a, for any equation in the form ax²+bx+c=0 where a is not zero. It always gives you both roots (solutions) of the equation in one calculation, regardless of whether the equation factors neatly.
What does the discriminant tell you?
The discriminant is the part under the square root, b²−4ac. If it's positive, the equation has two distinct real roots. If it's exactly zero, there's one repeated real root (a double root). If it's negative, the equation has no real roots — only two complex roots, since you'd be taking the square root of a negative number.
Do I always need the quadratic formula, or can I factor instead?
Factoring is faster when it works cleanly, but only some quadratics factor into whole numbers. The quadratic formula works for every quadratic equation without exception, including ones with ugly or irrational roots, which is why it's the fallback method taught alongside factoring.
What's the difference between a root and a solution?
They mean the same thing in this context — a root is a value of x that makes the equation equal zero, and solving the equation means finding all of its roots. A quadratic has at most two roots, which is a direct consequence of it being a second-degree (squared) equation.