Maths Lab · Free lesson · 24 minutes
Read a quadratic through its roots
Derive the sum and product of roots from factor expansion, form a quadratic and use root relationships in a transfer problem.
SS2 · quadratic structure01 · Think first
Start with a question.
You know the two numbers that make a quadratic zero. Can you reconstruct the equation without guessing its middle term? Expansion shows exactly where that term comes from.
What you already know
- Expand two brackets
- Apply the zero-product principle
- Recognise ax² + bx + c with a ≠ 0
02 · See it
Expand the root factors
The signs come from expansion. A quadratic with roots α and β is a(x − α)(x − β), where a is non-zero.
- Root factors
- a(x − α)(x − β)
- Expanded
- ax² − a(α + β)x + aαβ
- Compare coefficients
- α + β = −b/a; αβ = c/a
Use the justified worked steps below to explore this idea.
03 · Show steps · explain why
One justified step at a time.
- 1
(x − α)(x − β) = x² − (α + β)x + αβ
The two cross terms combine as −αx − βx. Their negative sum is the coefficient of x.
04 · Mistake clinic
Find the first wrong idea.
For x² − 7x + 10 = 0, the sum of roots is −7.
Compare the correction and its reason
The sum is −b/a = −(−7)/1 = 7.
The coefficient of x is the negative of the root sum. The factors (x − 5)(x − 2) expand to x² − 7x + 10.
Check: Expand the factors or substitute both roots. A sign rule should be supported by the structure.
A mistake is evidence about the next idea to repair. It is not a label for the learner.
05 · Your turn
Try first. Then use a hint.
06 · Check & reflect → next step
Understanding includes explaining.
Explain the negative sign in the root sum using bracket expansion, not a memorised formula alone.
Check a quadratic by factorisation and expansion before progressing to a different senior topic.
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JEM-S2-M04-001 · Phase 4B SS2 General Mathematics · Lesson 5 · printed pp. 20–21
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