Quadratic Equation
Solve ax² + bx + c = 0.
About the Quadratic Equation
Solve any equation of the form ax squared plus bx plus c equals zero, and see whether it has two real roots, one, or none.
How to use it
- Enter a, the coefficient of x squared.
- Enter b and c.
- Read the roots and the discriminant.
The formula
A quadratic is any equation where the highest power is two. The formula solves every one of them, which is unusual — most equations have no such universal key.
ax² + bx + c = 0 → x = [−b ± √(b² − 4ac)] ÷ 2a
The part under the root, b² − 4ac, is called the discriminant, and it tells you what kind of answer to expect before you finish the arithmetic:
positive → two real roots · zero → one repeated root · negative → two complex roots
Graphically the equation is a parabola, and the roots are where it crosses the x-axis. A negative discriminant means the curve never reaches the axis, which is why no real solution exists — not a failure of the method, but a fact about the shape.
Worked examples
| Equation | a, b, c | Discriminant | Roots |
|---|---|---|---|
| x² − 5x + 6 = 0 | 1, −5, 6 | 1 | x = 3 and x = 2 |
| 2x² + 4x − 6 = 0 | 2, 4, −6 | 64 | x = 1 and x = −3 |
| x² − 4x + 4 = 0 | 1, −4, 4 | 0 | x = 2 (repeated) |
| x² + 2x + 5 = 0 | 1, 2, 5 | −16 | complex: −1 ± 2i |
Checking a solution is easier than finding one, and always worth doing. For the first row, the roots should multiply to c/a and add to −b/a: 3 × 2 = 6 and 3 + 2 = 5. Both hold, so the answer is right. That relationship — Vieta's formulas — also makes factoring faster, because you are looking for two numbers with a known product and a known sum.
Common mistakes
- Losing the sign of b. In x² − 5x + 6, b is −5, so −b is +5. Dropping that sign is the single most common error in the whole formula, and it produces roots with the wrong sign.
- Getting b² wrong when b is negative. (−5)² is +25, not −25. A squared quantity is never negative, and a negative discriminant arriving from this step is usually this mistake.
- Dividing only part of the numerator by 2a. The entire −b ± √(...) is divided by 2a, not just the root. Writing the fraction with a full-width bar rather than a slash prevents this.
- Forgetting to rearrange to equal zero. The formula assumes ax² + bx + c = 0. An equation like x² = 5x − 6 must first become x² − 5x + 6 = 0, or every coefficient you read off is wrong.
Terms explained
- Quadratic
- An equation whose highest power is two. Its graph is always a parabola.
- Coefficient
- The numbers a, b and c multiplying x², x and the constant term. Read them with their signs.
- Discriminant
- b² − 4ac. Determines whether the roots are two, one or complex, before you solve anything.
- Root
- A value of x making the equation zero. Where the parabola crosses the x-axis.
- Parabola
- The U-shaped curve of a quadratic. It opens upward when a is positive and downward when a is negative.
- Vertex
- The turning point of the parabola, at x = −b ÷ 2a — exactly halfway between the two roots.
Common questions
- What does the discriminant tell me?
- Whether there are two real roots (positive), one repeated root (zero), or two complex roots (negative). It is worth computing first, because it tells you what to expect.
- What if the discriminant is negative?
- There is no real solution — the parabola never touches the x-axis. The complex roots still exist and matter in engineering, where they describe oscillation rather than a crossing point.
- Should I factor instead of using the formula?
- Factor if you can spot it quickly, since it is faster and less error-prone. The formula always works, so it is the reliable fallback when the factors are not obvious or do not exist in whole numbers.
- How do I check my answer?
- Substitute each root back and confirm you get zero. Or use Vieta's shortcut: the roots should add to −b/a and multiply to c/a.
- What is completing the square?
- An alternative method that rewrites the equation as a perfect square plus a constant. The quadratic formula is derived by completing the square on the general form, so they are the same technique at different levels of abstraction.
- Where do quadratics come up in practice?
- Anywhere something accelerates or an area is involved: projectile paths, braking distances, profit-maximising prices, and any problem where doubling one dimension quadruples a result.
- How do I find the highest or lowest point?
- The vertex sits at x = −b ÷ 2a. Substitute that back to get the value there. It is the maximum if a is negative and the minimum if a is positive.
- Can a be zero?
- No — then there is no x² term and the equation is linear, solved simply as bx + c = 0. The formula would also require dividing by zero.