Quadratic and Cubic Equation Solver: Roots, Discriminant, Vertex and Factorised Form

How this calculator works

Choose quadratic or cubic and enter the coefficients a, b, c (and d for a cubic).

For a quadratic you get the discriminant, the roots, the factorised form, the vertex and the sum and product of the roots.

For a cubic you get the three roots, real or complex, with a check.

Formulas used

• x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}
• \Delta = b^2 - 4ac
• x_v = -\frac{b}{2a}
• r_1 + r_2 = -\frac{b}{a},\ r_1r_2 = \frac{c}{a}

Worked example

With the default numbers in the calculator above:

1. Discriminant: 1.00 (positive: two different real roots)
2. Roots by the quadratic formula: x = 3.00 and x = 2.00
3. Factorised form: 1 (x - 3) (x - 2)
4. Vertex (the turning point) and axis of symmetry: (2.50, -0.25), axis x = 2.50, the curve opens upward (minimum)
5. Sum and product of the roots: 5.00 and 6.00
6. Vertex form: 1 (x - 2.5)^2 - 0.25

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Equation Cubic: ax^3 + bx^2 + cx + d = 0; a 1; b -6; c 11; d (cubic only) -6.

1. Depressed cubic t^3 + pt + q = 0 with x = t - b/(3a): p = -1.00, q = 0.00
2. Discriminant of the cubic: 4.00 (three different real roots)
3. The three roots: 3.00; 2.00; 1.00
4. Check: the equation evaluated at each real root: The largest value is 8.8818e-16 (it should be zero)
5. Sum and product of the roots: 6.00 and 6.00

Common questions

Related calculators

• Complex Roots Calculator: All Roots of a Complex Number
• Arithmetic and Geometric Sequence Calculator: nth Term, Sum of n Terms and Sum to Infinity
• Simultaneous Equations Solver (Matrix Method, 2 to 8 Unknowns)

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any school or first-year mathematics text (quadratic and cubic equations).

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