Quadratic formula calculator
Solve ax² + bx + c = 0, with the discriminant and every step
Type the three numbers a, b and c. The example is x² − 5x + 6 = 0.
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- Discriminant b² − 4ac
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- Turning point (vertex)
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- Factored form
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Working, step by step
How the equation is solved
Any equation of the form ax² + bx + c = 0, with a not zero, is solved by the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. Write the equation with everything on one side first. For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6. Take care with signs: if the equation says − 5x, b is −5.
The discriminant decides
The part under the square root, b² − 4ac, is called the discriminant. If it is more than 0 there are two different answers; if it is 0 there is one (the curve just touches the x-axis); if it is less than 0 there are no real answers, and the page gives the two complex answers instead. For the example, 25 − 24 = 1, so there are two answers: (5 + 1) ÷ 2 = 3 and (5 − 1) ÷ 2 = 2.
Exact answers
When the discriminant is a perfect square the answers are exact fractions, and the page also gives the factored form: x² − 5x + 6 = (x − 3)(x − 2). When it is not, the square root is simplified where possible (√8 = 2√2) and the decimals are rounded to 10 significant digits.
Turning point
The curve y = ax² + bx + c turns at x = −b ÷ 2a. For the example that is x = 2.5, where y = −0.25. Plot it on the graphing calculator to see the roots and the turning point.
When a is 0
Then the equation is a straight line, bx + c = 0, and the page solves that instead: x = −c ÷ b.
Formula and sources
- The quadratic formula and the discriminant: Common Core HSA-REI.B.4b.
- Checked in test_maths.js: x² − 5x + 6 = 0 gives 2 and 3; x² − 2x + 1 = 0 gives 1 (repeated); x² + 1 = 0 gives ±i; x² − 2 = 0 gives ±√2; 2x² + 3x − 2 = 0 gives 1/2 and −2.