Solving quadratic equations
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Solve it by factorisation when the factors are easy to spot, otherwise by the quadratic formula. The roots are also linked to the coefficients: sum of roots = −b/a and product of roots = c/a, which lets you build an equation from its roots.
Quadratic inequalities
To solve a quadratic inequality, first find the roots, then decide which region satisfies the sign. For an upward parabola, the expression is negative between the roots and positive outside them; for a downward parabola the two regions swap. A quick sketch or number line avoids sign errors. Take care with the boundary: use ≤ or ≥ to include the roots, and strict < or > to exclude them. If the leading coefficient is negative, either multiply through by −1 (remembering to reverse the inequality sign) or sketch the downward curve directly before reading off the solution set.
Key formula
Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Sum of roots = −b/a; product of roots = c/a. Equation from roots: x² − (sum)x + (product) = 0.
Worked example
Solve x² − 5x + 6 = 0 and then x² − 5x + 6 > 0.
Factorise: (x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3.
For the inequality, the parabola opens upward, so it is positive outside the roots: x < 2 or x > 3. (Between 2 and 3 the expression is negative.)
Remember
- Move all terms to one side so the equation equals 0 before factorising.
- For ax² + bx + c < 0 (a > 0) the solution lies between the roots.
- Always factorise or use the formula — never divide by x, as that loses a root.