AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Solving equations and inequalities: solve quadratic equations algebraically by factorising find approximate solutions using a graph
A quadratic equation has an x² term.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
A quadratic equation has an x² term. To solve algebraically by factorising, rearrange to the form ax² + bx + c = 0, then factorise into two brackets. For x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5: 2 and 3, so (x + 2)(x + 3) = 0. Setting each bracket to zero gives x = -2 or x = -3. If the coefficient of x² is not 1, factor out common factors or use appropriate factor pairs. To find approximate solutions using a graph, plot or read the curve y = ax² + bx + c and identify where it crosses the x-axis; those x-intercepts are the approximate roots. For example, from a graph the curve may cross near x = -2.1 and x = 0.6, so solutions are approximately -2.1 and 0.6. Always check by substitution.
What earns the marks
- Rearranges a quadratic equation into the standard form ax² + bx + c = 0 before factorising.
- Factorises a quadratic expression correctly, including cases where a common factor is taken out first.
- Uses the zero product property: if (x + p)(x + q) = 0 then x = -p or x = -q.
- Solves quadratic equations where the coefficient of x² is greater than 1 by factorising.
- Reads approximate solutions from the x-intercepts of the graph of y = ax² + bx + c.
- Checks algebraic solutions by substituting back into the original equation.
Common mistakes and how to put them right
Forgetting to rearrange to = 0 before factorising.
Put it right: First add 6 to both sides to get x² + 5x + 6 = 0, then factorise and set each bracket to 0.
Sign errors when finding the roots from brackets.
Put it right: Solve x + 2 = 0 to get x = -2 and x + 3 = 0 to get x = -3.
Reading the y-intercept instead of the x-intercepts from a graph.
Put it right: Approximate solutions are the x-values where the curve crosses the x-axis, not where it crosses the y-axis.
Exam tips
- Always rearrange the equation to the form ax² + bx + c = 0 before factorising, and show this step clearly.
- When using a graph, state that you are reading the x-axis intercepts and give your answers to a suitable degree of accuracy, such as one decimal place.
- Check each solution by substituting it back into the original equation; this catches sign errors quickly.
Try the lesson
The start of the Solving equations and inequalities: solve quadratic equations algebraically by factorising find approximate solutions using a graph lesson, made from checked teaching notes for this specification point.
Have a go first
Solve x² + 7x + 12 = 0 by factorising.
Listen
A quadratic equation has an x² term. To solve algebraically by factorising, rearrange to the form ax² + bx + c = 0, then factorise into two brackets. For x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5: 2 and 3, so (x + 2)(x + 3) = 0. Setting each bracket to zero gives x = -2 or x = -3. If the coefficient of x² is not 1, factor out common factors or use appropriate factor pairs. To find approximate solutions using a graph, plot or read the curve y = ax² + bx + c and identify where it crosses the x-axis; those x-intercepts are the approximate roots. For example, from a graph the curve may cross near x = -2.1 and x = 0.6, so solutions are approximately -2.1 and 0.6. Always check by substitution. A strong answer earns marks for these points. Rearranges a quadratic equation into the standard form ax² + bx + c = 0 before factorising. Factorises a quadratic expression correctly, including cases where a common factor is taken out first. Uses the zero product property: if (x + p)(x + q) = 0 then x = -p or x = -q. Solves quadratic equations where the coefficient of x² is greater than 1 by factorising. A common slip: Forgetting to rearrange to = 0 before factorising; for example, solving x² + 5x = -6 by writing (x + 2)(x + 3) = -6 and then incorrectly setting each bracket to -6. Instead, first add 6 to both sides to get x² + 5x + 6 = 0, then factorise and set each bracket to 0. Exam tip: Always rearrange the equation to the form ax² + bx + c = 0 before factorising, and show this step clearly.
See it
Key words
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Factorising quadratic expressions
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Using graphs to find approximate roots
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Checking solutions by substitution
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Algebraic manipulation and rearrangement
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See every definition
- Factorising quadratic expressions
- Rearranging a quadratic expression into two brackets, such as finding two numbers that multiply to the constant and add to the coefficient of x.
- Using graphs to find approximate roots
- Plotting or reading the curve y = ax² + bx + c and identifying where it crosses the x-axis, as those x-intercepts are the approximate roots.
- Checking solutions by substitution
- Substituting each solution back into the original equation to catch errors such as sign mistakes.
- Algebraic manipulation and rearrangement
- Rearranging a quadratic equation to the form ax² + bx + c = 0 before factorising.
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