Algebra and Graphs: Solving quadratic equations (factorising, quadratic formula, completing the square)

What you'll learn

Quadratic equations appear in virtually every CIE IGCSE Mathematics examination paper, worth significant marks across both Paper 2 (Extended) and Paper 4. This revision guide covers the three essential methods for solving quadratic equations: factorising, using the quadratic formula, and completing the square. Understanding when and how to apply each method is crucial for examination success.

Key terms and definitions

Quadratic equation — an equation of the form ax² + bx + c = 0, where a ≠ 0, and a, b, and c are constants.

Factorising — rewriting a quadratic expression as the product of two linear factors, typically in the form (px + q)(rx + s).

Roots — the solutions to a quadratic equation; the values of x that make the equation equal to zero (also called solutions or zeros).

Discriminant — the expression b² - 4ac from the quadratic formula, which determines the nature and number of roots.

Completing the square — rewriting a quadratic expression in the form a(x + p)² + q to solve equations or find turning points.

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

Zero product property — if the product of two factors equals zero, then at least one of the factors must equal zero; fundamental to solving factorised equations.

Core concepts

Factorising quadratic equations

Factorising remains the fastest method when applicable, particularly for simple quadratics where a = 1. CIE examiners expect candidates to recognise factorisable quadratics immediately.

When the coefficient of x² is 1 (x² + bx + c = 0):

1. Find two numbers that multiply to give c and add to give b
2. Write the equation as (x + p)(x + q) = 0
3. Apply the zero product property: x + p = 0 or x + q = 0
4. Solve each linear equation

Example: Solve x² + 7x + 12 = 0

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

When the coefficient of x² is not 1 (ax² + bx + c = 0):

Two approaches exist: the 'ac method' (splitting the middle term) or inspection.

ac method:

1. Multiply a and c
2. Find two numbers that multiply to ac and add to b
3. Split the middle term using these numbers
4. Factorise by grouping

Example: Solve 2x² + 7x + 3 = 0

• ac = 2 × 3 = 6
• Numbers that multiply to 6 and add to 7: 6 and 1
• 2x² + 6x + 1x + 3 = 0
• 2x(x + 3) + 1(x + 3) = 0
• (2x + 1)(x + 3) = 0
• x = -½ or x = -3

Difference of two squares: Recognise the pattern a² - b² = (a + b)(a - b). Questions testing x² - k = 0 appear frequently in CIE papers.

The quadratic formula

The quadratic formula provides solutions for any quadratic equation, making it essential when factorising is difficult or impossible. For ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / (2a)

Step-by-step application:

1. Rearrange the equation into standard form ax² + bx + c = 0
2. Identify the values of a, b, and c (watch signs carefully)
3. Substitute into the formula
4. Simplify the discriminant (b² - 4ac)
5. Calculate both solutions using + and -
6. Simplify answers or leave in surd form as required

The discriminant (b² - 4ac) determines:

• b² - 4ac > 0: two distinct real roots
• b² - 4ac = 0: one repeated real root (equal roots)
• b² - 4ac < 0: no real roots

CIE IGCSE Extended papers regularly test candidates' ability to use the discriminant to determine the number of roots without solving completely.

Example: Solve 3x² - 5x - 2 = 0 using the quadratic formula

• a = 3, b = -5, c = -2
• x = (5 ± √(25 - 4(3)(-2))) / (2 × 3)
• x = (5 ± √(25 + 24)) / 6
• x = (5 ± √49) / 6
• x = (5 ± 7) / 6
• x = 12/6 = 2 or x = -2/6 = -⅓

Completing the square

Completing the square transforms a quadratic into the form (x + p)² + q, revealing the minimum/maximum point and enabling solution of equations. This method appears in both algebra questions and coordinate geometry contexts.

For x² + bx + c:

1. Write x² + bx as (x + b/2)²
2. Remember (x + b/2)² expands to x² + bx + (b/2)²
3. Subtract (b/2)² and add c to maintain equality
4. Final form: (x + b/2)² + (c - (b/2)²)

Example: Write x² + 6x + 2 in the form (x + p)² + q

• Half of 6 is 3
• (x + 3)² = x² + 6x + 9
• x² + 6x + 2 = (x + 3)² - 9 + 2
• x² + 6x + 2 = (x + 3)² - 7

To solve by completing the square:

1. Complete the square to reach (x + p)² + q = 0
2. Rearrange to (x + p)² = -q
3. Take square roots: x + p = ±√(-q)
4. Solve: x = -p ±√(-q)

Example: Solve x² + 8x - 5 = 0 by completing the square

• (x + 4)² - 16 - 5 = 0
• (x + 4)² - 21 = 0
• (x + 4)² = 21
• x + 4 = ±√21
• x = -4 ± √21

For ax² + bx + c where a ≠ 1:

1. Factor out the coefficient a from the x² and x terms only
2. Complete the square inside the bracket
3. Expand the factor a back through

Example: Write 2x² + 12x + 7 in completed square form

• 2(x² + 6x) + 7
• 2[(x + 3)² - 9] + 7
• 2(x + 3)² - 18 + 7
• 2(x + 3)² - 11

Choosing the appropriate method

CIE examinations sometimes specify which method to use, but often candidates must select strategically:

Use factorising when:

• The question specifies "by factorising"
• The quadratic looks simple (small integer coefficients)
• You need the fastest method under time pressure

Use the quadratic formula when:

• The question specifies "using the formula"
• Factorising is not immediately obvious
• Answers are likely to be non-integer or involve surds
• The discriminant is mentioned

Use completing the square when:

• The question explicitly requires this method
• Finding the turning point of a parabola
• Writing in the form (x + p)² + q is requested
• Sketching graphs or identifying transformations

Solving equations requiring algebraic manipulation

CIE papers frequently present quadratics that require rearrangement first:

Equations not in standard form:

• Expand brackets: (x + 3)(x - 2) = 10 becomes x² + x - 6 = 10, then x² + x - 16 = 0
• Collect terms: 5x² = 3x + 2 becomes 5x² - 3x - 2 = 0
• Clear fractions: multiply through by the lowest common denominator

Non-linear simultaneous equations:

When one equation is linear (y = mx + c) and one quadratic (y = ax² + bx + c), substitute the linear into the quadratic to eliminate y.

Worked examples

Example 1: Solve the equation 6x² + 11x - 10 = 0

Solution:

Using factorisation (ac method):

• ac = 6 × (-10) = -60
• Numbers multiplying to -60 and adding to 11: 15 and -4
• 6x² + 15x - 4x - 10 = 0
• 3x(2x + 5) - 2(2x + 5) = 0
• (3x - 2)(2x + 5) = 0
• 3x - 2 = 0 or 2x + 5 = 0
• x = ⅔ or x = -2.5 [2 marks]

Example 2: A rectangle has length (x + 7) cm and width (x + 2) cm. The area of the rectangle is 78 cm².

(a) Form an equation in x and show that it simplifies to x² + 9x - 64 = 0 [2 marks]

(b) Solve the equation to find the value of x [3 marks]

Solution:

(a) Area = length × width

• (x + 7)(x + 2) = 78
• x² + 2x + 7x + 14 = 78
• x² + 9x + 14 = 78
• x² + 9x - 64 = 0 ✓ [2 marks]

(b) Using the quadratic formula where a = 1, b = 9, c = -64:

• x = (-9 ± √(81 - 4(1)(-64))) / 2
• x = (-9 ± √(81 + 256)) / 2
• x = (-9 ± √337) / 2
• x = (-9 + 18.36) / 2 or x = (-9 - 18.36) / 2
• x = 4.68 or x = -13.68
• Since x represents a length dimension, x = 4.68 cm (reject negative) [3 marks]

Example 3: By completing the square, solve x² - 10x + 7 = 0, giving your answers in the form p ± q√2 [4 marks]

Solution:

• x² - 10x + 7 = 0
• (x - 5)² - 25 + 7 = 0 [completing the square]
• (x - 5)² - 18 = 0
• (x - 5)² = 18
• x - 5 = ±√18
• x - 5 = ±√(9 × 2)
• x - 5 = ±3√2
• x = 5 ± 3√2 [4 marks]

Common mistakes and how to avoid them

• Mistake: Forgetting to include both roots when solving. Writing only x = 3 when factorising gives (x - 3)(x + 2) = 0. Correction: Always solve each factor separately. Here, x - 3 = 0 gives x = 3 AND x + 2 = 0 gives x = -2. State both solutions.

• Mistake: Sign errors when identifying a, b, c in the quadratic formula, particularly when b or c is negative. For x² - 5x + 2 = 0, using b = 5 instead of b = -5. Correction: Write out a = ..., b = ..., c = ... explicitly before substituting. Include the sign with the number.

• Mistake: Forgetting to divide by 2a in the quadratic formula, only dividing by a or by 2. Correction: The entire numerator (-b ± √(b² - 4ac)) must be divided by 2a. Use brackets in your calculator: (numerator) ÷ (2a).

• Mistake: When completing the square, forgetting to subtract (b/2)² after adding it implicitly. Writing x² + 6x = (x + 3)² instead of (x + 3)² - 9. Correction: Remember (x + 3)² = x² + 6x + 9, so x² + 6x = (x + 3)² - 9. Always subtract the squared term.

• Mistake: Not rearranging to standard form before applying methods. Attempting to factorise x² + 5x = 14 directly. Correction: Always rearrange to ax² + bx + c = 0 first: x² + 5x - 14 = 0, then factorise to (x + 7)(x - 2) = 0.

• Mistake: Accepting impossible solutions in context questions. Giving x = -8 for a length measurement. Correction: Read the question context. Reject negative values for lengths, ages, or quantities. State clearly which solution is rejected and why.

Exam technique for solving quadratic equations

• Command words matter: "Solve by factorising" requires factorisation with clear working shown. "Solve" without specification allows any valid method, but choose strategically. "Use the formula" or "Complete the square" demand specific approaches—using another method scores zero marks.

• Show all working: CIE mark schemes award method marks even when final answers are incorrect. Write out the formula, substitutions, and each algebraic step. Particularly for 3-4 mark questions, examiners look for process marks: forming the equation (1 mark), correct substitution (1 mark), correct method (1 mark), correct solutions (1 mark).

• Surd and exact answers: Questions stating "give your answer in exact form" or "in surd form" require √ notation, not decimals. Simplify surds: √18 = 3√2. Questions asking for answers "correct to 2 decimal places" require calculator evaluation of surds.

• Context questions: Form the equation, simplify fully as instructed, solve using the appropriate method, then interpret solutions in context. Typically worth 5-7 marks total across multiple parts, with marks allocated to formation (1-2), simplification (1), solving (2-3), and interpretation (1).

Quick revision summary

Quadratic equations ax² + bx + c = 0 have three solution methods. Factorise when coefficients are simple integers, using (px + q)(rx + s) = 0. Apply the quadratic formula x = (-b ± √(b² - 4ac))/(2a) for all equations, especially when surds appear. Complete the square into (x + p)² + q form for turning points or when specified. Always check the discriminant determines root count: positive gives two roots, zero gives one, negative gives none. Choose methods strategically and show complete working for maximum marks.