Quadratic equations are the cornerstone of secondary school mathematics across GCSE, IGCSE, A-Level, and IB curricula. They appear in projectile motion, area calculations, revenue optimization, and geometry proofs.

While many students can memorize the quadratic formula, confusion arises when deciding which method to use under exam conditions, or when dealing with tricky negative signs and coefficients greater than 1. Below is the complete step-by-step masterclass with worked examples.

1. What is a Quadratic Equation? (Standard Form)

A quadratic equation is a second-degree polynomial equation whose highest exponent is 2, written in standard form as ax² + bx + c = 0, where a ≠ 0.

Before you touch any method, your first step must always be to rearrange the equation so that all terms are on one side, set equal to zero:

ax^2 + bx + c = 0

For example, if you see 3x^2 = 5x + 2, subtract 5x and 2 from both sides to obtain standard form: 3x^2 - 5x - 2 = 0, where a = 3, b = -5, and c = -2.

2. Method 1: Factoring (Factorisation)

Use factoring when a, b, and c are integers that easily break into two binomial brackets (x + p)(x + q) = 0, relying on the Zero Product Property.

The Zero Product Property states: if A \times B = 0, then either A = 0 or B = 0.

Worked Example: Factoring when a = 1

Solve: x^2 - 7x + 12 = 0

1
Find two numbers that multiply to give +12 and add to give -7. The numbers are -3 and -4 (since -3 \times -4 = +12 and -3 + -4 = -7).
2
Write as two brackets: (x - 3)(x - 4) = 0.
3
Set each factor to zero:
x - 3 = 0 \implies x = 3
x - 4 = 0 \implies x = 4

When a > 1 (Factoring by Grouping): For 2x^2 + 7x + 3 = 0, multiply a \times c = 2 \times 3 = 6. Find factors of 6 that add to 7: 6 and 1. Split the middle term: 2x^2 + 6x + 1x + 3 = 0. Factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0, giving roots x = -1/2 and x = -3.

3. Method 2: The Quadratic Formula & Discriminant Analysis

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any quadratic equation; the discriminant b² - 4ac dictates whether there are 2, 1, or 0 real solutions.

The universal formula derived from completing the square is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Discriminant Value (\Delta = b^2 - 4ac) Nature of Roots Geometric Meaning (Graph of y = ax^2+bx+c)
b^2 - 4ac > 0 Two distinct real roots The parabola crosses the x-axis at two distinct points.
b^2 - 4ac = 0 One repeated real root The vertex of the parabola touches the x-axis (tangent line).
b^2 - 4ac < 0 No real roots (two complex roots) The parabola lies entirely above or below the x-axis with no x-intercepts.
Worked Example: The Quadratic Formula

Solve to 2 decimal places: 2x^2 - 5x - 4 = 0

1
Identify coefficients: a = 2, b = -5, c = -4.
2
Substitute into formula with careful brackets around negatives:
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-4)}}{2(2)}
3
Simplify the discriminant: (-5)^2 - 4(2)(-4) = 25 - (-32) = 57.
x = \frac{5 \pm \sqrt{57}}{4}
4
Evaluate both roots:
x_1 = \frac{5 + 7.5498}{4} \approx 3.14
x_2 = \frac{5 - 7.5498}{4} \approx -0.64

4. Method 3: Completing the Square (Vertex Form)

Completing the square rewrites ax² + bx + c into a(x + p)² + q, immediately revealing the turning point coordinates (-p, q) and line of symmetry.

For an equation where a = 1: x^2 + bx = (x + b/2)^2 - (b/2)^2.

Worked Example: Completing the Square

Solve: x^2 + 6x - 7 = 0

1
Take half of the coefficient of x: half of 6 is 3.
2
Write the square bracket and subtract the square of 3: (x + 3)^2 - 3^2 - 7 = 0.
3
Simplify constants: (x + 3)^2 - 9 - 7 = (x + 3)^2 - 16 = 0. (The minimum turning point is at (-3, -16)).
4
Isolate the square and take square roots:
(x + 3)^2 = 16 \implies x + 3 = \pm \sqrt{16} = \pm 4
x = -3 + 4 = 1 or x = -3 - 4 = -7.

5. Exam Decision Matrix: Which Method in 5 Seconds?

Check if factors are obvious; if not, or if the question specifies 'give your answer to 3 significant figures', deploy the quadratic formula immediately.
  • If the question says "Give answers to 2 decimal places / 3 significant figures": Do not waste time attempting to factorise. Deploy the Quadratic Formula immediately.
  • If the question says "Find the coordinates of the turning point": Use Completing the Square.
  • If the question carries only 2 or 3 marks and integer terms are small: Test Factoring first.

Frequently Asked Questions: Quadratic Equations

The four main methods to solve quadratic equations are: (1) Factoring (quickest when integer factors exist), (2) The Quadratic Formula (universal for any equation), (3) Completing the Square (essential for finding turning points and derivations), and (4) Graphical solutions (finding the x-intercepts of the parabola where y = 0).
The discriminant is the expression under the square root in the quadratic formula. If b² - 4ac > 0, the equation has two distinct real roots. If b² - 4ac = 0, the equation has one repeated real root (the parabola is tangent to the x-axis). If b² - 4ac < 0, there are no real roots (the roots are complex/imaginary, and the parabola never crosses the x-axis).
Use completing the square when the question explicitly instructs you to write the expression in vertex form a(x + p)² + q, when asked to find the maximum or minimum point of a quadratic curve, or when the coefficient of x² is 1 and the coefficient of x is an even integer (making division by 2 clean).
To factorise ax² + bx + c when a > 1, find two numbers that multiply to give ac and add to give b. Then split the middle linear term bx into two terms using these numbers, and factorise by grouping the first two terms and the last two terms.

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