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6
4x = 24, so x = 6.
A Level · English Medium · Pearson Edexcel · Mathematics
Equations, patterns and functions
Solve equations and interpret a function’s rule, roots and rate of change.
About this lesson and its coverage
Shared cross-level core and stretch unit. Start with the core questions, then use the stretch example for deeper work. The same unit is available in related class and board pathways; it is not a full grade-specific or examination-specification course. A teacher should select the appropriate depth and further topics.
Learn the idea
An equation is true at its solution values. Keep both sides balanced when rearranging. A function assigns one output to each allowed input. A root is an input making the output zero; a graph can show roots and changing rates.
Worked example
3x + 4 = 19 gives 3x = 15, so x = 5. For f(x) = x² − 5x + 6 = (x − 2)(x − 3), the roots are 2 and 3.
- State any restrictions on inputs.
- Use reversible algebraic steps.
- Substitute the result back to check.
Stretch example
For f(x) = x³ − 3x, the derivative f′(x) = 3x² − 3 gives stationary inputs x = −1 and 1. The second derivative is 6x: negative at −1 gives a local maximum; positive at 1 gives a local minimum.
Your written practice
Show your working on paper. Check your answer, then choose Next.
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17
2 × 9 − 1 = 17.
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17
The constant difference is 4.
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3
(x − 3)(x − 4) = 0.
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Division by zero is undefined.
State the domain restriction rather than assigning an infinite value.
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2
−1 + 3 = 2.
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The integral is [x²] from 1 to 3 = 9 − 1 = 8 square units.
Use an antiderivative and both bounds; the curve is nonnegative on this interval.
Progress and browser storage
Notebook challenge, sources and further study
One function, three views
Represent a quadratic as a formula, value table and sketch.
Review with a guardian or teacher
- Label the axes and domain used.
- Check roots by substitution.
- Explain how the three representations agree.
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