Algebra: Equations, Inequalities and Simultaneous Equations

This lesson covers solving quadratic and simultaneous equations, including the case of one linear and one quadratic, and representing inequality solutions on graphs.

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Solving Quadratic Equations

By Factorising

Set equal to zero, factorise, then solve each factor.

x² − 7x + 10 = 0 → (x − 2)(x − 5) = 0 → x = 2 or x = 5

Using the Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a (see previous lesson).

By Completing the Square

Convert to (x + p)² = q form, then square root both sides (see previous lesson).

Solving Equations Reducible to Quadratics

Look for a substitution. For x⁴ − 5x² + 4 = 0, let u = x²: u² − 5u + 4 = 0 → (u − 1)(u − 4) = 0 → u = 1 or u = 4 → x = ±1 or x = ±2

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Linear Simultaneous Equations

Elimination Method

Example: 3x + 2y = 12 and 5x − 2y = 4 Add the equations: 8x = 16 → x = 2 → y = 3