Read zero information without solving first
For ax^2+bx+c, two compact relations reveal the sum and product of zeroes directly from coefficients.
Learning goals
- Find quadratic zeroes by factorisation.
- Use and verify the sum and product relations.
- Solve expressions and parameter conditions involving the zeroes.
Concept lesson
Factorisation bridge
If p(x)=(x-alpha)(x-beta), then alpha and beta make a factor zero and are the zeroes.
The two quadratic relations
For p(x)=ax^2+bx+c, a not equal to zero, alpha+beta=-b/a and alpha beta=c/a. The sum alone has a minus sign; both denominators are the leading coefficient a.
Find zeroes by factorisation
Find the zeroes of 2x^2-7x+3.
Verify both relations
For 2x^2-7x+3 with zeroes 1/2 and 3, verify the relations.
Find alpha squared plus beta squared
If alpha and beta are zeroes of x^2-5x+3, find alpha^2+beta^2 without solving.
Reciprocal zeroes determine a parameter
The zeroes of x^2+3x+k are reciprocals. Find k.
Sign-and-denominator check
For 3x^2+8x-4, first label a=3, b=8, c=-4. Complete alpha+beta=-__/__ and alpha beta=__/__. Say where each sign comes from.
Translate a relation before calculating
For 2x^2-8x+(k+1), one zero is seven times the other. Write beta=7 alpha, combine it with alpha+beta=4, and then use the product relation to find k.
Choose solve or relate
For each target—individual zeroes, their sum, alpha^2+beta^2—choose whether factorisation or coefficient relations are more efficient and justify the choice.
Three-mark verification structure
Show factorisation and both zeroes; then write two separate equalities comparing alpha+beta with -b/a and alpha beta with c/a. Numerical answers without both comparisons do not complete 'verify'.
Exit ticket
For 2x^2-5x-3: find and verify the zeroes; find alpha^2+beta^2 using relations. Then state what reciprocal zeroes would imply about c/a for any quadratic.