Quadratic Equation Solver

Give it the three coefficients of ax squared plus bx plus c and this finds where the equation equals zero. It tells you whether there are two real roots, one repeated root or two complex ones, and shows the discriminant, the vertex, which way the curve opens and the y-intercept. If a is zero, it notices and solves the equation as a straight line instead.

How to use it

1. Enter a, b and c.
2. Read the number and type of solutions.
3. Check the discriminant, vertex and direction.
4. Copy the roots.

When it helps

• Solving homework and exam practice.
• Finding where a curve crosses the x-axis.
• Locating the highest or lowest point of a parabola.
• Checking your own working.

Good to know

• A negative discriminant means no real solutions, only complex ones involving i.
• The vertex is the minimum if a is positive and the maximum if it's negative.
• Decimals work fine, so you can paste coefficients from a calculation.

The quadratic formula

For ax squared plus bx plus c equals zero, the solutions are x equals minus b plus or minus the square root of b squared minus 4ac, all divided by 2a. The part under the root, the discriminant, decides what kind of answers you get: positive gives two real roots, zero gives one, and negative gives two complex roots.

The tool applies the formula and also finds the vertex at x equals minus b over 2a, which is the turning point of the curve.

Worked example: a ball thrown upward

A ball's height is given by minus 5t squared plus 20t plus 1 metres. When does it hit the ground? Set height to zero with a of minus 5, b of 20 and c of 1. The roots are about minus 0.05 and 4.05 seconds. Time can't be negative, so the ball lands after about 4.05 seconds, and the vertex at t equals 2 gives a maximum height of 21 metres.

Quadratics model many things: projectile paths, areas, profit curves and bridge arches.

Factoring and checking

If the roots are whole numbers, the quadratic factors neatly. For x squared minus 3x plus 2, the roots are 1 and 2, so it factors as (x minus 1)(x minus 2). You can check any root by substituting it back: the result should be zero.

If the discriminant is a perfect square, the roots are rational. Otherwise they involve a square root and appear as decimals here.

Complex roots

When the discriminant is negative, the curve never crosses the x-axis and the roots are complex numbers, written as a real part plus or minus an imaginary part with i. For x squared plus 2x plus 5, the roots are minus 1 plus or minus 2i.

Complex roots always come in pairs for equations with real coefficients, and they matter in physics and engineering, for example in oscillating circuits.

Questions