Quadratic equation solver

Roots 2, 1

1x² + -3x + 2 · discriminant 1

What kind two real roots

Discriminant 1

First root 2

Second root 1

Vertex x 1.5

Vertex y -0.25

Discriminant b² − 4ac decides the roots

The discriminant tells you the answer before you solve

The b² − 4ac term under the square root decides everything. Positive gives two distinct real roots, zero gives one repeated root where the parabola just touches the axis, and negative gives a complex pair. Checking it first tells you what kind of answer to expect, and it is the fastest way to spot an arithmetic slip — a negative discriminant on a problem that should have real roots means a sign is wrong.

The quadratic formula gives x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 3x + 2, the discriminant is 1 and the roots are 2 and 1. A negative discriminant means the roots are a complex conjugate pair.

How to solve a quadratic

1 Enter the coefficients a, b and c, with signs.

2 Read the roots and the discriminant.

3 Check the vertex for the turning point of the parabola.

4 Note that a cannot be zero — that would be a linear equation.

Completing the square is where the formula comes from, and it also gives the vertex directly. Writing ax² + bx + c as a(x + b/2a)² + (c − b²/4a) shows the turning point at x = −b/2a immediately. That form is often more useful than the roots — for a projectile it gives the maximum height, and for an optimisation problem it gives the answer without needing calculus.

Questions

What is the quadratic formula? −

x = (−b ± √(b² − 4ac)) ÷ 2a.

What does the discriminant tell me? +

What if a is zero? +

Where is the vertex? +

Can I always factorise instead? +

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