The quadratic formula solves ax² + bx + c = 0 with x = (−b ± √D) ÷ 2a, where D = b² − 4ac is the discriminant. For x² − 3x − 4 = 0, D = 9 + 16 = 25, so x = (3 ± 5) ÷ 2 and the solutions are x = 4 and x = −1.
What is the quadratic formula?
The quadratic formula, known in the Netherlands as the abc-formule, gives the solutions of any equation of the form ax² + bx + c = 0 with a ≠ 0. It is x = (−b ± √(b² − 4ac)) ÷ (2a). The expression under the root, D = b² − 4ac, is the discriminant. It tells you how many solutions to expect: two real solutions if D > 0, one repeated solution if D = 0 and no real solutions if D < 0 (two complex ones instead).
Step by step
- Move everything to one side so the equation equals 0, then read off a, b and c. For 2x² + 5x = 3 this becomes 2x² + 5x − 3 = 0, so a = 2, b = 5, c = −3.
- Calculate D = b² − 4ac = 25 + 24 = 49.
- Take the square root: √49 = 7.
- Calculate x₁ = (−5 + 7) ÷ 4 = 0.5 and x₂ = (−5 − 7) ÷ 4 = −3.
The discriminant and the parabola
The graph of y = ax² + bx + c is a parabola and the solutions are where it crosses the x-axis. If a is positive the parabola opens upwards and has a minimum, if a is negative it opens downwards and has a maximum. The vertex lies at x = −b ÷ 2a, exactly halfway between the two solutions. The calculator also reports the y-intercept, which is always c.
When is factorising quicker?
Sometimes you can factorise by inspection: x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or x = 3. That works when you can find two numbers with sum −5 and product 6. When it does not come quickly, the quadratic formula always works. For equations without a constant term (ax² + bx = 0), take out a common factor: x(ax + b) = 0.
Links to other topics
- The eigenvalues of a 2×2 matrix solve the quadratic λ² − trace·λ + determinant = 0. See the Eigenvalue Calculator.
- Squares and roots also appear in the Pythagoras Calculator.
- For growth and ratios try the Percentage Calculator.
How to use this calculator
- Write the equation in the form ax² + bx + c = 0.
- Enter a, b and c, using a minus sign for negative numbers.
- Press Calculate to see the discriminant D and the number of solutions.
- Read the solutions and the vertex of the matching parabola.
Key figures at a glance
| Formula | x = (−b ± √D) ÷ 2a |
|---|---|
| Discriminant | D = b² − 4ac |
| D > 0 | two real solutions |
| D = 0 | one solution |
| D < 0 | no real solutions |
| Vertex | x = −b ÷ 2a |
Frequently asked questions
Calculate the discriminant D = b² − 4ac and put it in x = (−b ± √D) ÷ 2a. You get two solutions, or one if D = 0.
The equation has no real solutions: the parabola never crosses the x-axis. It has two complex solutions.
Then it is not quadratic but linear (bx + c = 0), with solution x = −c ÷ b.
The x-coordinate is −b ÷ 2a. Substitute it into the equation for the y-coordinate.
When factorising is not quick, when the solutions are not whole numbers or when the discriminant is negative.