A zero is an input that makes the polynomial equal to zero. For a quadratic, the sum and product of zeroes let you move between roots and coefficients. Keep the leading coefficient visible: shortcuts valid for a monic polynomial can fail when it is not monic.
Curriculum checked 2 October 2026 · original practice for selected skills.
You will need: Factorisation, substitution and multiplication of brackets.
Attempt each task before opening a hint. If you need help, reveal the first hint, then the setup, then compare your written steps with the solution. Finish with the retests without referring back.
For ax²+bx+c with a≠0, the sum of zeroes is −b/a and their product is c/a.
If zeroes are r and s, a monic quadratic with those zeroes is (x−r)(x−s). Non-zero scalar multiples have the same zeroes.
Substitution checks a claimed zero directly. A factor x−r corresponds to the zero r, not −r.
Find two numbers with sum 7 and product 12.
Use factors x−3 and x−4.
x²−7x+12=(x−3)(x−4).
3+4=7=−(−7)/1; 3 × 4=12.
Answer: 3 and 4
Before moving on: can you explain why your method works, as well as give the answer?
Write a method as well as an answer. The tasks change the reasoning, not just the numbers.
Here a is 2, not 1.
Use −b/a and c/a.
−(−5)/2=5/2.
3/2.
Answer: Sum 5/2; product 3/2
Before moving on: can you explain why your method works, as well as give the answer?
Use x−(−2) and x−5.
Expand (x+2)(x−5).
A zero of −2 gives x+2.
x²−5x+2x−10=x²−3x−10.
Answer: x²−3x−10
Before moving on: can you explain why your method works, as well as give the answer?
Substitute x=2.
4+2k−6=0.
2k=2 gives k=1.
The product is −6, so the other zero is −3.
Answer: k=1; other zero −3
Before moving on: can you explain why your method works, as well as give the answer?
A tempting claim: “The root sum is the coefficient of x.”
For a general quadratic it is −b/a, including both a sign change and a division.
These changed questions test whether you can reconstruct the method. If you use a hint, record where you got stuck and retry later. Completing this small set does not establish full chapter mastery.
Multiply the two linear factors.
(x−1)(x−6).
x²−6x−x+6.
The middle term is −7x.
Answer: x²−7x+6
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
The leading coefficient is 3.
Sum=−4/3; product=−2/3.
The negative sign in −b/a reverses +4.
Keep the negative constant.
Answer: Sum −4/3; product −2/3
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
If a retest exposed the same error, rewrite the first incorrect step and explain its correction aloud. If both were independent, return to a mixed exercise or a missed paper question. A parent can ask what changed in the method rather than only asking for the answer.
Not without an additional condition such as being monic. Multiplying a polynomial by a non-zero constant preserves its zeroes.
Yes. Use the two print buttons above. The question sheet leaves working space and hides solutions; the worked-solutions option includes the solution steps. Your browser can save either view as a PDF.
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