What is the quadratic formula?

The quadratic formula solves ax squared plus bx plus c equals zero. The roots are minus b, plus or minus the square root of b squared minus four a c, all over two a. The calculator applies it for you.

What is the discriminant?

The discriminant is b squared minus four a c. If it is positive there are two real roots, if zero there is one repeated root, and if negative the roots are complex. The calculator reports which case applies.

If a is zero the equation is no longer quadratic but linear, and the quadratic formula does not apply. Enter a non zero value for a to solve a quadratic.

Quadratic Equation Calculator

Solve a quadratic equation of the form ax² + bx + c = 0. Enter the three coefficients and the calculator gives the roots using the quadratic formula, the discriminant, and the vertex of the parabola. It handles real and complex roots.

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x = 4, x = -1

roots

Discriminant0

Vertex0

How it works

The calculator works out the discriminant, b squared minus four a c, to see how many roots there are, then applies the quadratic formula. The vertex, the turning point of the parabola, is at x equals minus b over two a.

Worked example

For x squared minus 3x minus 4, the discriminant is 9 plus 16, which is 25. The roots are x equals 4 and x equals minus 1, and the vertex is at x equals 1.5.

Related calculators

Slope Intercept Form Calculator: straight lines.
Square Root Calculator: roots of a number.
Graphing Calculator: plot the parabola.
Scientific Calculator: advanced functions.

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