Quadratic equations
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. You can solve it by factorising, by completing the square, or with the quadratic formula.
Key idea
x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac gives the number of real roots: Δ > 0 two distinct roots, Δ = 0 one repeated root, Δ < 0 no real roots.
Quadratic inequalities
First find the roots, then decide which region satisfies the inequality. A quick sketch of the parabola helps. When a > 0 the curve is below the x-axis (negative) between the roots and above (positive) outside them.
Worked example
Solve x² − 5x + 6 > 0. Factorise: (x − 2)(x − 3) > 0, so the roots are 2 and 3. The parabola opens upward, so the expression is positive outside the roots. Answer: x < 2 or x > 3.
Modulus equations and inequalities
For |ax + b| = c with c ≥ 0, write ax + b = c or ax + b = −c and solve both. For |ax + b| < c, use −c < ax + b < c.
Remember
- |x| ≥ 0 always, so |x| = (a negative number) has no solution.
- Always check answers by substitution.
- |x| > k (k > 0) means x < −k or x > k.