Quadratic Equations

What is a Quadratic Equation?

A quadratic equation has the standard form:

ax² + bx + c = 0, where a ≠ 0

The graph of y = ax² + bx + c is a parabola — a U-shape (when a > 0) or an inverted U (when a < 0). Solving the equation means finding the x-values where the parabola crosses the x-axis (the "roots" or "zeros").

Quadratics can have 0, 1, or 2 real solutions depending on the discriminant.

Key Terms

• Root / Solution: an x-value that satisfies the equation (makes both sides equal)
• Parabola: the U-shaped curve of a quadratic graph
• Discriminant: b² - 4ac, which tells you how many real solutions exist
• Vertex: the turning point of the parabola (minimum or maximum)
• Completing the square: rewriting ax² + bx + c in the form a(x + p)² + q

Method 1: Factorising

Best when the equation factorises neatly into whole numbers.

Find two numbers that multiply to c and add to b, then write the factorised form.

Example: x² + 5x + 6 = 0

• Numbers that multiply to 6 and add to 5: 2 and 3
• Factorise: (x + 2)(x + 3) = 0
• Set each factor to zero: x + 2 = 0 or x + 3 = 0
• Solutions: x = -2 or x = -3

Example: x² - 7x + 12 = 0

• Numbers that multiply to 12 and add to -7: -3 and -4
• (x - 3)(x - 4) = 0
• Solutions: x = 3 or x = 4

Difference of two squares: x² - 9 = 0 → (x + 3)(x - 3) = 0 → x = 3 or x = -3

Method 2: The Quadratic Formula

Use this when factorising is difficult or impossible. It always works.

x = (-b ± √(b² - 4ac)) / 2a

Example: 2x² - 3x - 2 = 0 (a = 2, b = -3, c = -2)

• Discriminant: (-3)² - 4(2)(-2) = 9 + 16 = 25
• x = (3 ± √25) / 4 = (3 ± 5) / 4
• x = 8/4 = 2 or x = -2/4 = -0.5

Example: x² + 2x - 8 = 0 (a = 1, b = 2, c = -8)

• Discriminant: 4 - 4(1)(-8) = 4 + 32 = 36
• x = (-2 ± 6) / 2
• x = 4/2 = 2 or x = -8/2 = -4

Method 3: Completing the Square

Rewrite the quadratic in the form (x + p)² + q = 0, then solve by rearranging.

Example: x² + 6x + 2 = 0

• Take half the coefficient of x: half of 6 = 3
• Write: (x + 3)² - 9 + 2 = 0
• (x + 3)² - 7 = 0
• (x + 3)² = 7
• x + 3 = ±√7
• x = -3 + √7 or x = -3 - √7

Completing the square is also used to find the vertex of a parabola: vertex is at (-p, q).

The Discriminant

Δ = b² - 4ac determines the number of real solutions:

Example: 3x² - 12x + 12 = 0

• a = 3, b = -12, c = 12
• Δ = (-12)² - 4(3)(12) = 144 - 144 = 0 → one repeated solution

Choosing the Right Method

Common Mistakes to Avoid

• Forgetting the ± in the quadratic formula: there are two solutions when Δ > 0
• Sign errors when factorising: if c is positive and b is negative, both brackets contain minus signs
• Dropping the = 0: you must set each factor equal to zero, not just write the factorised form
• Errors in completing the square: subtracting p² inside is essential — (x + p)² = x² + 2px + p², so you must subtract p²

Tips and Tricks

• Always rearrange to ax² + bx + c = 0 before applying any method
• Check solutions by substituting back into the original equation
• If the question says "solve exactly," leave answers in surd form (e.g. x = -3 + √7)
• If the discriminant is a perfect square, the quadratic will factorise — try factorising first