These eight original questions cover the whole quadratic functions and equations section. Work each on paper first, then compare your working with the answer.
Questions 1 and 2 test recognition, 3 to 5 test solving, and 6 to 8 test graphs and context.
Practice questions
Question 1. Which of these is quadratic after simplifying? (A) x(x − 3) + 4 (B) (x + 1)² − x² − 2x (C) 3 − 5x².
Answer
(A) expands to x² − 3x + 4, so it is quadratic.
(B) expands to x² + 2x + 1 − x² − 2x = 1, a constant.
(C) is quadratic with a = −5, b = 0, c = 3. So (A) and (C) are quadratic.
Question 2. Write 2x² + 9 = 7x in general form and state a, b and c.
Answer
Subtract 7x from both sides: 2x² − 7x + 9 = 0.
So a = 2, b = −7, c = 9. The sign of b stays with the coefficient.
Question 3. Solve x² + 3x − 18 = 0.
Answer
Two numbers that multiply to −18 and add to 3 are 6 and −3. So (x + 6)(x − 3) = 0.
The roots are x = −6 and x = 3. Check x = 3: 9 + 9 − 18 = 0.
Question 4. Solve x² = 8x − 15.
Answer
Move everything to one side: x² − 8x + 15 = 0. The numbers −3 and −5 multiply to 15 and add to −8.
So (x − 3)(x − 5) = 0, and x = 3 or x = 5. A student who writes x(x − 8) = −15 has no way to continue.
Question 5. Solve 2x² − 5x − 3 = 0.
Answer
Multiply a and c: 2 × (−3) = −6. Two numbers that multiply to −6 and add to −5 are −6 and 1.
Split: 2x² − 6x + x − 3 = 0, so 2x(x − 3) + 1(x − 3) = 0, giving (2x + 1)(x − 3) = 0.
The roots are x = −½ and x = 3. Check x = 3: 18 − 15 − 3 = 0.
Question 6. The function y = x² − 2x − 8 has the table: y = 7 at x = −3, 0 at x = −2, −5 at x = −1, −8 at x = 0, −9 at x = 1, −8 at x = 2, −5 at x = 3, 0 at x = 4 and 7 at x = 5. Use it to state the roots of x² − 2x − 8 = 0, the turning point, and the solutions of x² − 2x − 8 = −5.
Answer
y = 0 at x = −2 and x = 4, so the roots are −2 and 4.
The lowest point is (1, −9), and 1 is halfway between −2 and 4.
For y = −5, the table gives x = −1 and x = 3, so the solutions are x = −1 and x = 3.
Question 7. From the graph in question 6, a student says “the roots are −8 and −9”. What has the student read, and what is the correct reading?
Answer
The student has read y-values. The value −8 is the y-intercept, and −9 is the minimum value.
Roots are x-values where y = 0, so the correct roots are x = −2 and x = 4.
Question 8. A stall owner models daily profit as P = −x² + 12x − 20 in RM hundreds, where x is the price in RM. Find the price that gives the maximum profit and state that profit.
Answer
Factorise −(x² − 12x + 20) = −(x − 2)(x − 10), so the roots are 2 and 10. The turning point has x = (2 + 10) ÷ 2 = 6.
P = −36 + 72 − 20 = 16. The coefficient of x² is negative, so this is a maximum.
The maximum profit is RM1 600 (16 hundred) when the price is RM6.
If you got some wrong
Wrong answers to questions 1 and 2 point back to recognising a quadratic from its highest power. Questions 3 to 5 link to solving factorisable quadratic equations.
Questions 6 and 7 are covered in finding roots from a quadratic graph, and question 8 in interpreting a turning point in a contextual problem.
To build a timed session from mixed topics, use the timed practice session builder. For a teacher to look at your working, see online one-to-one Mathematics tuition.