Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0: enter a, b and c to get the discriminant, both roots (real or complex), the vertex of the parabola and the steps of the quadratic formula.
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Solve ax² + bx + c = 0
Type a minus sign for negative values, such as −5 or -5.
How do I solve a quadratic equation?
Use the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac decides the roots: positive gives two real roots, zero one repeated root, negative a complex pair. For x² − 5x + 6 = 0 it is 1, so x = (5 ± 1) ÷ 2: 3 or 2.
- Write the equation as ax² + bx + c = 0 and enter a, b and c. Include minus signs, such as −5.
- Read the discriminant and the roots: two real numbers, one repeated root, or a complex pair.
- Check the vertex of the parabola and the worked steps of the quadratic formula.
Formulas & Methods
Quadratic formula
Example: x² − 5x + 6: x = (5 ± 1) ÷ 2 = 3 or 2
Discriminant
Example: x² + 2x + 5: D = 4 − 20 = −16, complex roots −1 ± 2i
Vertex
Example: x² − 5x + 6: (2.5, −0.25)
Key features
Real or complex roots
Two real roots, a repeated root, or a complex pair written as a ± bi.
Worked steps
The discriminant and the quadratic formula with your numbers, so you can follow or check the working.
Vertex of the parabola
The turning point of y = ax² + bx + c, useful for graphs and maximum or minimum problems.
When a is 0
With a = 0 the equation isn’t quadratic but linear, bx + c = 0, with the single solution x = −c ÷ b. The solver says so and gives that root. If b is 0 too, the equation either has no solution (c isn’t 0) or is true for every x (c is 0).
Frequently Asked Questions
What does a negative discriminant mean?
The parabola never crosses the x-axis, so there are no real solutions. The two solutions are complex numbers, a ± bi, where i is the square root of −1.
Can I factorise instead of using the formula?
When the roots are whole numbers or simple fractions, yes: x² − 5x + 6 = (x − 2)(x − 3). The formula always works, including for roots like √2 that don’t factorise neatly.
Can the coefficients be decimals or fractions?
Decimals, yes, such as 0.5. For a fraction, enter its decimal value, or multiply the whole equation by the denominator first: ½x² + x = 0 becomes x² + 2x = 0.
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