AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Solving equations and inequalities: including those that require rearrangement including completing the square and by using the quadratic formula (Higher tier only)
Higher tier students solve quadratic equations that need rearrangement before solving.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Higher tier students solve quadratic equations that need rearrangement before solving. Rearrange to ax² + bx + c = 0, then choose a method. Completing the square writes x² + bx + c as (x + b/2)² - (b/2)² + c; for example, x² + 6x + 5 = 0 becomes (x + 3)² - 4 = 0, so (x + 3)² = 4, x + 3 = ±2, giving x = -1 or x = -5. The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any quadratic in standard form; for 2x² + 3x - 2 = 0, a = 2, b = 3, c = -2, so x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4, giving x = 0.5 or x = -2. Both methods may produce surd solutions; leave them exact unless asked for decimals.
What earns the marks
- Rearranges a quadratic equation into the form ax² + bx + c = 0 before completing the square or using the quadratic formula.
- Completes the square correctly, including when the coefficient of x² is not 1, by factoring out that coefficient first.
- Solves the equation obtained after completing the square, remembering both the positive and negative square roots.
- Substitutes the correct values of a, b and c into the quadratic formula, including negative signs.
- Simplifies the resulting expression, leaving surd answers in exact form where required.
Common mistakes and how to put them right
Forgetting to rearrange to = 0 before using the quadratic formula.
Put it right: Rearrange to ax² + bx + c = 0 and identify a, b and c from that form.
Sign errors when substituting negative values into the quadratic formula.
Put it right: Write down a, b and c clearly, then substitute carefully, using brackets for negative numbers.
When completing the square, forgetting to subtract (b/2)² after adding it, or losing the constant term.
Put it right: Check the expansion of your completed-square form to confirm it matches the original expression.
Exam tips
- If a question asks for solutions in surd form, leave the answer with a square root and simplify it fully; do not round to a decimal.
- When using the quadratic formula, write down the values of a, b and c before substituting to reduce sign errors.
- After completing the square, show the step where you take the square root of both sides, including the ± sign, to secure method marks.
Try the lesson
The start of the Solving equations and inequalities: including those that require rearrangement including completing the square and by using the quadratic formula (Higher tier only) lesson, made from checked teaching notes for this specification point.
Have a go first
When solving the equation x² + 6x + 5 = 0 by completing the square, which equation represents a correct intermediate step?
Listen
Higher tier students solve quadratic equations that need rearrangement before solving. Rearrange to ax² + bx + c = 0, then choose a method. Completing the square writes x² + bx + c as (x + b/2)² - (b/2)² + c; for example, x² + 6x + 5 = 0 becomes (x + 3)² - 4 = 0, so (x + 3)² = 4, x + 3 = ±2, giving x = -1 or x = -5. The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any quadratic in standard form; for 2x² + 3x - 2 = 0, a = 2, b = 3, c = -2, so x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4, giving x = 0.5 or x = -2. Both methods may produce surd solutions; leave them exact unless asked for decimals. A strong answer earns marks for these points. Rearranges a quadratic equation into the form ax² + bx + c = 0 before completing the square or using the quadratic formula. Completes the square correctly, including when the coefficient of x² is not 1, by factoring out that coefficient first. Solves the equation obtained after completing the square, remembering both the positive and negative square roots. Substitutes the correct values of a, b and c into the quadratic formula, including negative signs. A common slip: Forgetting to rearrange to = 0 before using the quadratic formula; for example, using the formula on x² + 3x = 4 without first writing x² + 3x - 4 = 0. Instead, rearrange to ax² + bx + c = 0 and identify a, b and c from that form. Exam tip: If a question asks for solutions in surd form, leave the answer with a square root and simplify it fully; do not round to a decimal.
See it
Key words
Say what each word means before you turn the card.
Rearranging quadratic equations
Try answering aloud or in your head before turning the card.
Completing the square
Try answering aloud or in your head before turning the card.
Using the quadratic formula
Try answering aloud or in your head before turning the card.
Exact surd solutions
Try answering aloud or in your head before turning the card.
See every definition
- Rearranging quadratic equations
- Solving quadratic equations that need rearrangement before solving, by rearranging to ax² + bx + c = 0 and then choosing a method.
- Completing the square
- Writes x² + bx + c as (x + b/2)² - (b/2)² + c, including when the coefficient of x² is not 1, by factoring out that coefficient first.
- Using the quadratic formula
- The formula x = (-b ± √(b² - 4ac)) / (2a) solves any quadratic in standard form, substituting the correct values of a, b and c including negative signs.
- Exact surd solutions
- Both methods may produce surd solutions; leave them exact unless asked for decimals, with the answer left with a square root and simplified fully.
Label the diagram
Some boxes have lost their names. Pick the right name for each number.
1Box 1
2Box 2
3Box 3
There is more to this lesson
The full lesson goes on to check your understanding and help you earn the marks. Create a free account to carry on, with your progress saved.
A week of Premium from your first lesson · Then the free plan continues · No card, no charge
Test your understanding of Solving equations and inequalities: including those that require rearrangement including completing the square and by using the quadratic formula (Higher tier only)
Active recall beats re-reading. Take an instant quiz now to test what you’ve learned and lock it into long-term memory.
A week of Premium from your first lesson · Then the free plan continues · No card, no charge