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

This calculator solves any quadratic equation of the form ax² + bx + c = 0, the standard form you meet in NCEA maths, university papers and problems involving areas, projectile motion or optimisation. You enter the three coefficients, a, b and c, and the tool instantly applies the quadratic formula to find both roots, the x values where the parabola crosses zero. Alongside the roots it works out the discriminant, b squared minus four a c, which shows what kind of solution you have: a positive discriminant gives two distinct real roots, zero gives one repeated root, and a negative discriminant means the roots are complex, shown with the real and imaginary parts. The calculator also returns the vertex of the parabola, the turning point at x equals minus b over two a, useful for sketching the curve or finding a maximum or minimum value. Results update live as you change any coefficient, so you can experiment with different equations and see straight away how the roots, discriminant and vertex shift. Remember that a cannot be zero, because that turns the equation into a straight line rather than a quadratic, and the calculator will flag this if you try it. Whether you are checking homework, studying for an exam or working through a real-world problem that reduces to a quadratic, this tool gives you the full working in seconds.

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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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