Solving quadratic equations

Factorisation is efficient when the factors are simple. Otherwise completing the square or the quadratic formula works for every quadratic ax² + bx + c = 0 with a ≠ 0. The discriminant b² − 4ac tells you whether the real roots are distinct, repeated or absent.

If an equation originally contains fractions, denominators must remain nonzero. Substituting each candidate root into the original equation checks both the algebra and these restrictions. Keep roots exact unless the question asks for a decimal approximation.

Worked example

Solve 2x² + 3x − 2 = 0.

1. Factorise: (2x − 1)(x + 2) = 0.
2. At least one factor is zero.
3. Solve 2x − 1 = 0 or x + 2 = 0.

Answer: x = 1/2 or x = −2

Revise first: Expanding and factorising.