Quadratic equations and the quadratic formula

What you'll learn

Quadratic equations appear extensively across CIE IGCSE Mathematics papers, particularly in Paper 2 and Paper 4 (Extended tier). This topic covers solving quadratic equations by factorisation, completing the square, and using the quadratic formula, plus interpreting solutions in real-world contexts. Understanding when to apply each method and handling both integer and non-integer solutions forms a crucial part of your exam technique.

Key terms and definitions

Quadratic equation — an equation that can be written in the form ax² + bx + c = 0, where a ≠ 0 and a, b, c are constants.

Roots (or solutions) — the values of x that satisfy the quadratic equation; graphically, these are the x-intercepts of the parabola.

Discriminant — the expression b² - 4ac, which determines the nature of the roots (two distinct real roots, one repeated root, or no real roots).

Completing the square — rewriting a quadratic expression in the form a(x + p)² + q to facilitate solving or identify the turning point.

Quadratic formula — the formula x = (-b ± √(b² - 4ac))/(2a) used to solve any quadratic equation ax² + bx + c = 0.

Factorisation — expressing a quadratic as a product of two linear factors, typically in the form (px + q)(rx + s) = 0.

Coefficient — the numerical factor of a term; in ax² + bx + c, the coefficient of x² is a, of x is b, and c is the constant term.

Null factor law — the principle that if AB = 0, then A = 0 or B = 0, used when solving factorised quadratic equations.

Core concepts

Standard form of quadratic equations

Every quadratic equation must be arranged in standard form ax² + bx + c = 0 before applying solution methods. This means:

All terms moved to one side of the equation
The right-hand side equals zero
Terms written in descending powers of x

Examples of rearranging:

3x² = 5x - 2 becomes 3x² - 5x + 2 = 0
x(x + 4) = 12 expands to x² + 4x - 12 = 0
(2x - 1)² = 9 expands to 4x² - 4x + 1 = 9, then 4x² - 4x - 8 = 0

Solving by factorisation

Factorisation works efficiently when the quadratic can be expressed as a product of two linear factors. For equations where a = 1:

Find two numbers that multiply to give c and add to give b
Write as (x + p)(x + q) = 0
Apply the null factor law: x + p = 0 or x + q = 0
Solve each linear equation

For x² + 5x + 6 = 0:

Numbers that multiply to 6 and add to 5: 2 and 3
(x + 2)(x + 3) = 0
x = -2 or x = -3

When a ≠ 1, use the cross-multiplication method or factor by grouping:

For 2x² + 7x + 3 = 0:

Find factors of 2 × 3 = 6 that add to 7: these are 6 and 1
Rewrite: 2x² + 6x + x + 3 = 0
Group: 2x(x + 3) + 1(x + 3) = 0
Factor: (2x + 1)(x + 3) = 0
Solutions: x = -½ or x = -3

Always check whether the quadratic is a difference of two squares (a² - b² = (a + b)(a - b)):

x² - 25 = 0 gives (x + 5)(x - 5) = 0, so x = ±5
4x² - 9 = 0 gives (2x + 3)(2x - 3) = 0, so x = ±3/2

The quadratic formula

The quadratic formula solves any quadratic equation, particularly when factorisation proves difficult or impossible:

x = (-b ± √(b² - 4ac))/(2a)

Step-by-step method:

Write the equation in standard form ax² + bx + c = 0
Identify the values of a, b, and c (watch signs carefully)
Substitute into the formula
Calculate the discriminant b² - 4ac first
Evaluate both solutions using + and - signs
Simplify or round as required by the question

Critical points for CIE IGCSE Mathematics exams:

The formula is provided on the formula sheet, but you must identify a, b, c correctly
Negative values of b require careful bracketing: if b = -5, then -b = 5
Always calculate b² - 4ac separately before taking the square root
Show all working steps for method marks
Round final answers to 3 significant figures unless told otherwise

Completing the square

Completing the square transforms ax² + bx + c into the form a(x + p)² + q. This method reveals the turning point of the parabola at (-p, q) and provides an alternative solving technique.

For equations where a = 1, the process is:

Take half the coefficient of x: b/2
Square this value: (b/2)²
Write as (x + b/2)² - (b/2)² + c
Simplify the constants

Example: Complete the square for x² + 6x - 4

Half of 6 is 3
(x + 3)² - 9 - 4
(x + 3)² - 13

To solve (x + 3)² - 13 = 0:

(x + 3)² = 13
x + 3 = ±√13
x = -3 ± √13

When a ≠ 1, factor out the coefficient of x² first:

For 2x² + 8x + 5 = 0:

Factor: 2(x² + 4x) + 5 = 0
Complete inside brackets: 2[(x + 2)² - 4] + 5 = 0
Expand: 2(x + 2)² - 8 + 5 = 0
Simplify: 2(x + 2)² - 3 = 0

CIE IGCSE Mathematics papers frequently ask you to "express in the form a(x + p)² + q" — this tests completing the square without necessarily solving.

The discriminant and nature of roots

The discriminant Δ = b² - 4ac determines how many real solutions exist:

Δ > 0: two distinct real roots (parabola crosses x-axis twice)
Δ = 0: one repeated root (parabola touches x-axis at vertex)
Δ < 0: no real roots (parabola does not intersect x-axis)

Extended tier papers may ask you to:

Find conditions on a parameter for a quadratic to have two real roots
Prove a quadratic has no real solutions
Determine the relationship between coefficients for a repeated root

Example: For kx² + 4x + 1 = 0 to have two distinct real roots:

b² - 4ac > 0
16 - 4(k)(1) > 0
16 - 4k > 0
k < 4

Choosing the appropriate method

Exam efficiency requires selecting the optimal solution strategy:

Use factorisation when:

Integer solutions are expected
The question says "solve by factorisation"
You quickly spot factors
Time is limited and you recognise a pattern

Use the quadratic formula when:

Factorisation isn't obvious within 20-30 seconds
Non-integer solutions are expected
The question asks for answers to a certain number of significant figures
You need guaranteed accuracy

Use completing the square when:

The question explicitly requests it
You need to find the vertex/turning point
The quadratic is nearly a perfect square
Working with algebraic forms rather than numerical answers

Worked examples

Example 1: Factorisation method

Question: Solve the equation 3x² - 10x - 8 = 0

Solution: Find factors of 3 × (-8) = -24 that add to -10: these are -12 and 2

3x² - 12x + 2x - 8 = 0

Group the terms: 3x(x - 4) + 2(x - 4) = 0

Factor: (3x + 2)(x - 4) = 0

Apply null factor law: 3x + 2 = 0 or x - 4 = 0

x = -2/3 or x = 4

Example 2: Quadratic formula

Question: Solve 2x² + 5x - 1 = 0, giving your answers correct to 3 significant figures.

Solution: Identify coefficients: a = 2, b = 5, c = -1

Substitute into x = (-b ± √(b² - 4ac))/(2a):

x = (-5 ± √(25 - 4(2)(-1)))/(2 × 2)

x = (-5 ± √(25 + 8))/4

x = (-5 ± √33)/4

x = (-5 + 5.745)/4 or x = (-5 - 5.745)/4

x = 0.745/4 or x = -10.745/4

x = 0.186 or x = -2.69 (to 3 s.f.)

Example 3: Completing the square

Question: (a) Express x² - 8x + 5 in the form (x - p)² + q, where p and q are integers. (b) Hence solve x² - 8x + 5 = 0, leaving your answer in surd form.

Solution: (a) Half of -8 is -4 (x - 4)² - 16 + 5 (x - 4)² - 11

Therefore p = 4, q = -11

(b) (x - 4)² - 11 = 0 (x - 4)² = 11 x - 4 = ±√11

x = 4 + √11 or x = 4 - √11

Common mistakes and how to avoid them

• Forgetting to rearrange to standard form before solving — Students attempt to factorise or apply the formula without moving all terms to one side. Always ensure the equation equals zero first.

• Sign errors when identifying b in the quadratic formula — When the equation is x² - 5x + 6 = 0, b = -5, not +5. In the formula, -b becomes -(-5) = +5. Use brackets around negative values.

• Only finding one solution — Many students write x = 3 when both x = 3 and x = -2 satisfy the equation. The "±" symbol gives two answers; always state both unless the context eliminates one.

• Incorrect substitution in the quadratic formula — Errors often occur with 4ac when a or c is negative. For 2x² + 3x - 5 = 0, the discriminant is 9 - 4(2)(-5) = 9 + 40 = 49, not 9 - 40.

• Rounding too early — Students round √33 to 5.7 then divide, losing accuracy. Always keep full calculator values until the final step, then round to the specified number of significant figures or decimal places.

• Errors in completing the square with negative x coefficients — For x² - 6x + 2, students write (x - 3)² + 2 instead of (x - 3)² - 9 + 2. Remember to subtract the squared value (b/2)² before adding the constant term.

Exam technique for Quadratic equations and the quadratic formula

• Command words determine the method: "Factorise and solve" requires factorisation; "solve, giving your answers to 2 decimal places" indicates the quadratic formula; "express in the form..." explicitly requires completing the square. Read the question carefully before selecting your approach.

• Show systematic working for method marks: CIE IGCSE Mathematics mark schemes award marks for correct method even if the final answer is wrong. Write out your factorisation steps, or show b² - 4ac calculated separately, or display each stage of completing the square.

• Check solutions in context questions: If a quadratic models a real situation (dimensions, time, profit), negative or unrealistic solutions may need to be rejected. Always state which solution is valid and explain why the other is rejected, e.g., "time cannot be negative" or "length must be positive."

• Use the formula sheet strategically: The quadratic formula appears on the formula sheet provided in the exam, but you must identify a, b, and c independently. Write these values down explicitly in your working to avoid substitution errors and demonstrate clarity to examiners.

Quick revision summary

Quadratic equations ax² + bx + c = 0 can be solved by factorisation (for integer roots), the quadratic formula (for any equation), or completing the square (especially when finding turning points). Always rearrange to standard form first. The discriminant b² - 4ac determines the number of real roots: positive gives two, zero gives one repeated root, negative gives none. Choose your method based on the question wording and whether integer solutions are likely. Show all working steps for maximum method marks, and remember to state both solutions where they exist.