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GCSE MATHEMATICS — QUADRATIC EQUATIONS FORMULA BANK & REVISION
SHEET
Target: GCSE High School Exam Prep | Specification: AQA / Edexcel
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1. THE STANDARD FORM
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* A quadratic equation always takes the structural form:
ax² + bx + c = 0
* Crucial Condition: 'a' can never equal 0, otherwise the x² term
disappears and it becomes a basic linear equation.
* Roots: Every quadratic equation has exactly two answers (or roots).
2. METHOD 1: FACTORISING (Splitting the Middle Term)
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* Use this method if the numbers are simple and easy to break down.
* Goal: Find two numbers that multiply to give (a * c) and add up to
give the middle coefficient (b).
* Step-by-Step Worked Example: Solve x² - 5x + 6 = 0
- We need two numbers that multiply to +6 and add to -5.
- Those numbers are -2 and -3.
- Rewrite: x² - 2x - 3x + 6 = 0
- Factorise by grouping: x(x - 2) - 3(x - 2) = 0
- Final bracket form: (x - 2)(x - 3) = 0
- Therefore, the roots are: x = 2 and x = 3
3. METHOD 2: THE QUADRATIC FORMULA
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* Use this method if the quadratic expression does not factorise easily
or if the exam question tells you to "give your answer to 2 decimal places."
* The Universal Formula:
x = [ -b ± √(b² - 4ac) ] / 2a
4. THE DISCRIMINANT (The Nature of Roots)
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The specific value under the square root symbol, D = b² - 4ac, is called
the Discriminant. It tells you what type of roots you will get before
you even finish the math:
* If b² - 4ac > 0 (Positive number):
- You get Two Distinct Real Roots (two completely different answers).
* If b² - 4ac = 0 (Exactly zero):
- You get One Repeated Real Root (both answers are identical).
* If b² - 4ac < 0 (Negative number):
- You get No Real Roots (you cannot take the square root of a negative
number in standard GCSE math).
GCSE MATHEMATICS — QUADRATIC EQUATIONS FORMULA BANK & REVISION
SHEET
Target: GCSE High School Exam Prep | Specification: AQA / Edexcel
======================================================================
1. THE STANDARD FORM
----------------------------------------------------------------------
* A quadratic equation always takes the structural form:
ax² + bx + c = 0
* Crucial Condition: 'a' can never equal 0, otherwise the x² term
disappears and it becomes a basic linear equation.
* Roots: Every quadratic equation has exactly two answers (or roots).
2. METHOD 1: FACTORISING (Splitting the Middle Term)
----------------------------------------------------------------------
* Use this method if the numbers are simple and easy to break down.
* Goal: Find two numbers that multiply to give (a * c) and add up to
give the middle coefficient (b).
* Step-by-Step Worked Example: Solve x² - 5x + 6 = 0
- We need two numbers that multiply to +6 and add to -5.
- Those numbers are -2 and -3.
- Rewrite: x² - 2x - 3x + 6 = 0
- Factorise by grouping: x(x - 2) - 3(x - 2) = 0
- Final bracket form: (x - 2)(x - 3) = 0
- Therefore, the roots are: x = 2 and x = 3
3. METHOD 2: THE QUADRATIC FORMULA
----------------------------------------------------------------------
* Use this method if the quadratic expression does not factorise easily
or if the exam question tells you to "give your answer to 2 decimal places."
* The Universal Formula:
x = [ -b ± √(b² - 4ac) ] / 2a
4. THE DISCRIMINANT (The Nature of Roots)
----------------------------------------------------------------------
The specific value under the square root symbol, D = b² - 4ac, is called
the Discriminant. It tells you what type of roots you will get before
you even finish the math:
* If b² - 4ac > 0 (Positive number):
- You get Two Distinct Real Roots (two completely different answers).
* If b² - 4ac = 0 (Exactly zero):
- You get One Repeated Real Root (both answers are identical).
* If b² - 4ac < 0 (Negative number):
- You get No Real Roots (you cannot take the square root of a negative
number in standard GCSE math).
