Every equation that can be written as ax² + bx + c = 0 can be solved by this one formula, whether or not it factorises. Factorising is quicker when it works; the formula always works, which is why it is worth being able to use without thinking.
Reading the coefficients off correctly
| Symbol | Meaning |
|---|---|
| a | The number multiplying x². If it is zero the equation is not quadratic at all. |
| b | The number multiplying x, including its sign. |
| c | The number on its own, again including its sign. |
Rearrange the equation into the order ax² + bx + c = 0 before you read anything off. For 3x² = 7x − 2, move everything to one side first: 3x² − 7x + 2 = 0, giving a = 3, b = −7, c = 2. Reading b as 7 here — dropping the minus sign because it looks like a subtraction — is the single most common way this goes wrong.
Where a term is missing, its coefficient is zero. In x² − 9 = 0, b = 0. In 2x² + 5x = 0, c = 0.
The discriminant tells you the answer before you find it
The part under the square root, b² − 4ac, is called the discriminant, usually written D. Work it out first, because it tells you what kind of answer to expect:
| Discriminant | What it means |
|---|---|
| D > 0 | Two different real solutions. The curve crosses the x-axis twice. |
| D = 0 | One repeated solution. The curve just touches the x-axis. |
| D < 0 | No real solutions. The curve misses the x-axis entirely. At grades 11–12 the answers exist as a pair of complex numbers. |
Examiners ask about the discriminant on its own surprisingly often — “for what values of k does this equation have equal roots?” is really just “set b² − 4ac = 0 and solve for k”.
A perfect square discriminant is also a hint: it means the quadratic would have factorised, and you could have got there faster.
A worked example
Solve 2x² − 7x + 3 = 0.
Read off a = 2, b = −7, c = 3. Then the discriminant:
D = b² − 4ac = (−7)² − 4 × 2 × 3 = 49 − 24 = 25
25 is positive, so there are two real solutions, and it is a perfect square, so they will be tidy. Now the formula:
x = (−(−7) ± √25) ⁄ (2 × 2) = (7 ± 5) ⁄ 4
x₁ = (7 + 5) ⁄ 4 = 12 ⁄ 4 = 3
x₂ = (7 − 5) ⁄ 4 = 2 ⁄ 4 = 0.5
Now check, which takes ten seconds and is worth doing every time. The two roots should add to −b⁄a and multiply to c⁄a:
3 + 0.5 = 3.5, and −b⁄a = 7⁄2 = 3.5 ✓
3 × 0.5 = 1.5, and c⁄a = 3⁄2 = 1.5 ✓
Where marks get lost
Losing the sign of b. When b is negative, −b is positive. In the example above −(−7) became +7. Writing −7 there instead gives two wrong roots and no warning.
Dividing only part of the numerator. The 2a divides the whole of −b ± √D, not just the square root. Writing −b ± (√D ⁄ 2a) is a different formula and a different answer.
Squaring a negative b carelessly. (−7)² is 49, not −49. Brackets matter here.
Only giving one root. The ± means two answers unless the discriminant is zero. Unless the question restricts the domain — a length cannot be negative, a time before the start is not admissible — both are part of the answer.
Rounding mid-way. Keep the square root exact for as long as you can. √25 is 5 exactly; √26 is not, and rounding it to 5.1 before dividing will cost you accuracy marks.
Questions students actually ask
When should I factorise instead of using the formula?
Try factorising first when a is 1 and the numbers are small — it is faster and less error-prone. If nothing obvious appears within about twenty seconds, switch to the formula rather than staring at it.
What does it mean when the discriminant is negative?
There is no real number that solves the equation, and the parabola never crosses the x-axis. At grades 11–12 the solutions exist as a conjugate pair of complex numbers, which the calculator above will show you.
Why is the formula written with a ± sign?
A square root has two values: √25 is both +5 and −5. The ± captures both at once, which is why a quadratic normally has two solutions.
Can a quadratic have exactly one solution?
Yes, when the discriminant is exactly zero. Strictly it still has two solutions, they just happen to be the same number, which is why it is called a repeated or double root.
Where does the quadratic formula come from?
From completing the square on ax² + bx + c = 0 in general rather than with specific numbers. Doing that once produces the formula, which is why you never have to do it again.
Related formulas
Pythagoras’ theorem · Arithmetic progression · The equations of motion
Every calculation on this page runs inside your own browser. Nothing you type is sent anywhere, nothing is stored, and there is no account to make. If your school or college wants something built properly, tell us what you need.