From factors to roots
A root makes the quadratic equation true. Factorisation exposes the values that make one factor zero.
Learning goals
- Connect polynomial zeroes and equation roots.
- Factorise and apply the zero-product rule to find both roots.
- Verify roots in the original equation.
Concept lesson
Product ac and sum b
For ax^2+bx+c, seek two numbers whose product is ac and sum is b; split bx, group and factor.
Zero-product rule and root signs
If AB=0, then A=0 or B=0. Thus x-r=0 gives x=r, while x+r=0 gives x=-r.
Split and factorise
For 2x^2-5x+3=0, ac=6 and the pair -2,-3 sums to -5. Write 2x^2-2x-3x+3=0 and factor (2x-3)(x-1)=0.
Apply both factor equations
2x-3=0 gives x=3/2, and x-1=0 gives x=1. State both roots.
Read roots from factors
Write both roots of (x-4)(3x+2)=0. The scaffold requires each factor equation before its root.
Verify both roots
Substitute every found root into the original unfactorised equation. A non-zero result signals a factor or sign error.
Repair an incorrect factorisation
A student writes x^2+8x+12=(x+2)(x+4). Expand the proposed factors, locate the mismatch and repair the factorisation.
Two-mark factorisation answer
Show the split, grouping, two linear factors, zero-product step and both roots. Do not jump from the original equation to answers.
Exit ticket
Find the roots of 3x^2-7x+2=0 by factorisation and verify both roots in the original equation.