A reliable method
Set a quadratic equation equal to zero before factoring. Check a proposed sequence formula at the first few indices and inspect first and second differences.
The common trap
The roots are inputs where the expression equals zero, not the constant terms in the factors. Starting at n=1 differs from starting at n=0.
Make the connection
Use expansion to verify a factorisation and substitution to verify a root. These are different checks that catch different mistakes.
Check the middle term and the indexing
The equation x²+x−6=0 factors as (x+3)(x−2)=0, giving roots −3 and 2. The pair 3 and −2 has the required product −6 and sum 1. Reversing the signs changes the middle coefficient and solves another equation. Completing the square gives x²+6x+2=(x+3)²−7; the compensating constant must remove the extra nine. In a sequence beginning 3,7,11, the fifth term is 3+4(5−1)=19. Four transitions connect term one to term five. Always test a sequence rule at the first listed index.
Transfer, explain and check
For u_n=n²+2n, calculate u_4−u_3. The values 24 and 15 give 9. Compare (4−3)²+2(4−3)=3: substituting a difference is not generally the same as subtracting function outputs.
Find the larger root of x²−9x+18=0.
- Factor as (x−3)(x−6)=0.
- Set each factor to zero.
- Compare 3 and 6.
Compare the other answer choices
Return the smaller valid root despite the comparison word.
Report the sum of the roots instead of the larger root.
Use a factor constant rather than its zero.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
For x²−9x+18=0: (a) factor and give both roots; (b) identify the larger root; (c) if x and x−2 are rectangle lengths, decide whether either root must be rejected and justify.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Cambridge IGCSE Mathematics 0580