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General Solution of a Quadratic Equation is taught in the note for Quadratic Equations. Read it below and look for the section on this topic.

Quadratic Equations

11 min read

What is a quadratic equation?

A quadratic equation can be written as with . It usually has two solutions, called roots. Graphically the roots are where the curve crosses the x-axis.

The curve is a parabola. It meets the x-axis at x = 1 and x = 2: those are the roots.
A quadratic graphThe graph of y equals x squared minus 2x minus 3. It is a U-shaped curve called a parabola. It crosses the x axis at x equals minus 1 and x equals 3, has its lowest point, the vertex, at 1 and minus 4, and is symmetrical about the line x equals 1.xyaxis of symmetry x = 1(−1, 0)(3, 0)vertex (1, −4)(0, −3)
Another quadratic, y = x² − 2x − 3. The roots are where the curve crosses the x-axis; the lowest point is the vertex.

Solving by factorising

Worked exampleSolve x² − 5x + 6 = 0

Find the roots of .

  1. Two numbers that multiply to 6 and add to −5: −2 and −3.
  2. If two numbers multiply to zero, one of them is zero.

Answer: or .

The quadratic formula

Works for every quadratic equation.
  • the discriminant: it tells how many real roots there are
The discriminant D = b² − 4ac
DRootsGraph
D > 0two different real rootscrosses the x-axis twice
D = 0one repeated roottouches the x-axis
D < 0no real rootsdoes not reach the x-axis
Try itExplore ax² + bx + c

Change , and . Watch the curve, the discriminant and the roots. A dash means there is no real root.

Worked exampleUse the formula

Solve .

  1. a = 2, b = 3, c = −5.
  2. Substitute.
  3. Two roots.

Answer: or .

Check yourself What are the roots of x² − 9 = 0?
Check yourself x² + 2x + 5 = 0 has D = 4 − 20 = −16. What does that mean?

Key terms

Quadratic equation
An equation of the form ax² + bx + c = 0.
Root
A value of x that makes the equation true.
Discriminant
b² − 4ac; decides the number of real roots.
Parabola
The U-shaped graph of a quadratic.

Summary

  • Roots are where the parabola crosses the x-axis.
  • Factorise, then set each bracket to zero.
  • Formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
  • The discriminant tells whether there are two, one or no real roots.