Simultaneous equations and quadratic inequalities

Simultaneous equations

• Linear and non-linear: substitute the linear equation into the other. For y = x + 1 and x² + y² = 13: x² + (x + 1)² = 13 gives 2x² + 2x − 12 = 0, so x² + x − 6 = 0 and (x + 3)(x − 2) = 0. The solutions are x = 2, y = 3 and x = −3, y = −2.
• Three unknowns: eliminate one unknown from two pairs of equations, then solve the two equations that are left.
• Example: x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2. Adding the first and third gives 2x + 3y = 8; adding the second and third gives 3x + y = 5. Solving gives x = 1, y = 2 and z = 3.

Quadratics

• Solve by factorising, completing the square or the formula x = (−b ± √(b² − 4ac)) ÷ 2a.
• Completing the square: x² + 4x − 1 = 0 becomes (x + 2)² = 5, so x = −2 ± √5.
• The discriminant b² − 4ac tells you the number of real roots: positive means two, zero means one (repeated) and negative means none.

Quadratic inequalities

• Find the critical values by solving the equation, sketch the graph, then read off the region.
• x² − x − 6 < 0: (x − 3)(x + 2) < 0. The curve is below the x-axis between the roots, so −2 < x < 3.
• x² − x − 6 > 0 gives two regions: x < −2 or x > 3.
• x² ≥ 9 means x ≤ −3 or x ≥ 3. Don't just write x ≥ 3.

Key terms

Simultaneous equations

Equations that are true at the same time.

Critical values

The solutions of the equation used to solve an inequality.

Discriminant

b² − 4ac, which shows how many real roots a quadratic has.

Completing the square

Writing a quadratic in the form (x + p)² + q.

Quadratic inequality

An inequality involving x².