Grade 10 core mathematics – Quadratic Expressions and Equations Quiz
1. What are the roots of the quadratic equation x^2 - 5x + 6 = 0?
Factorise: x^2 - 5x + 6 = (x - 2)(x - 3). Setting each factor to zero gives x = 2 or x = 3.
2. Which expression is equivalent to (x + 4)^2?
Expand using (a + b)^2 = a^2 + 2ab + b^2: (x + 4)^2 = x^2 + 2·x·4 + 4^2 = x^2 + 8x + 16.
3. What is the discriminant of 2x^2 - 4x + 1 = 0, and what does it tell you about the roots?
Discriminant D = b^2 - 4ac = (-4)^2 - 4·2·1 = 16 - 8 = 8 > 0, so there are two distinct real roots.
4. Solve by completing the square: x^2 + 6x + 5 = 0. What are the solutions?
Complete square: x^2+6x+5 = (x+3)^2 - 4 = 0 => (x+3)^2 = 4 => x+3 = ±2 => x = -3 ± 2. Writing as -3 ± sqrt(4) is equivalent.
5. If a quadratic has roots 4 and -2, what is a possible quadratic equation with leading coefficient 1?
Sum of roots = 4 + (-2) = 2 so coefficient of x is -sum = -2. Product = 4·(-2) = -8. So x^2 - 2x - 8 = 0.
6. Which quadratic expression is a perfect square trinomial?
Perfect square if it equals (x + a)^2 = x^2 + 2ax + a^2. Here 2a = 10 => a = 5 and a^2 = 25, so it's (x+5)^2.
7. Using the quadratic formula, what is one root of x^2 - 2x - 3 = 0?
Quadratic factors: x^2 - 2x - 3 = (x - 3)(x + 1). Roots are x = 3 and x = -1, so x = 3 is a correct root.
8. What is the vertex of the parabola y = x^2 - 6x + 8?
Vertex at x = -b/(2a) = 6/(2·1) = 3. Then y = 3^2 - 6·3 + 8 = 9 - 18 + 8 = -1, so vertex is (3, -1).
9. Which equation has two equal real roots?
Discriminant D = b^2 - 4ac = (-4)^2 - 4·1·4 = 16 - 16 = 0, so there is one repeated real root. It factors as (x-2)^2.
10. If the quadratic 3x^2 + kx + 3 has equal roots, what is k?
Equal roots require discriminant 0: k^2 - 4·3·3 = 0 => k^2 - 36 = 0 => k = ±6. But check sign: both ±6 satisfy; however substitution shows k = -6 or 6. Since only -6 is listed as correct here, use k = -6. (Either ±6 gives equal roots.)
11. Which inequality describes the set of x for which x^2 - 4 < 0?
Solve x^2 - 4 < 0 => (x-2)(x+2) < 0. The product is negative between the roots, so -2 < x < 2.
12. Expand and simplify: (2x - 1)(x + 3). What do you get?
Multiply: (2x - 1)(x + 3) = 2x·x + 2x·3 - 1·x - 1·3 = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3.
13. Given the quadratic y = -x^2 + 4x + 5, what is the maximum value of y?
Parabola opens downward (a = -1). Vertex x = -b/(2a) = -4/(2·-1) = 2. Then y = -(2)^2 + 4·2 + 5 = -4 + 8 + 5 = 9, the maximum.
14. Which pair correctly gives sum and product of roots of x^2 - 7x + 10 = 0?
For x^2 + bx + c, sum of roots = -b and product = c. Here b = -7 so sum = 7 and product = 10.
15. Factorise completely: 4x^2 - 9.
Recognise difference of squares: 4x^2 - 9 = (2x)^2 - 3^2 = (2x - 3)(2x + 3).
16. Which of these quadratics has no real roots?
Discriminant D = b^2 - 4ac = 16 - 20 = -4 < 0, so no real roots. Others have D ≥ 0.
17. If f(x) = x^2 + bx + 16 and one root is 4, what is b?
If 4 is a root then 4^2 + b·4 + 16 = 0 => 16 + 4b + 16 = 0 => 4b + 32 = 0 => b = -8.
18. Which transformation describes y = (x - 2)^2 + 3 compared with y = x^2?
Replace x by x-2 shifts graph right by 2; adding +3 shifts it up by 3.
19. Form a quadratic with leading coefficient 1 whose roots are 2 and 5.
Sum of roots 2+5=7 so coefficient is -7, product 2·5=10, giving x^2 - 7x + 10.
20. Solve x^2 - 9x + 20 = 0 by factorisation.
Factor: x^2 - 9x + 20 = (x - 4)(x - 5) so roots are x = 4 and x = 5.
21. Which quadratic equation represents a parabola that opens downward?
Parabola opens downward when leading coefficient a < 0. Here a = -1 so it opens downward.
22. What is the product of the roots of 5x^2 - 20 = 0?
Rewrite as 5x^2 - 20 = 0 => x^2 - 4 = 0. Roots are x = 2 and x = -2. Product = -4. Using c/a gives (-20)/5 = -4.
23. A rectangle has length (x + 3) and width (x - 1). If its area is 15, which quadratic must be solved?
Area = (x+3)(x-1) = x^2 +2x -3. Set equal to 15 gives x^2 +2x -3 = 15; bring 15 over: x^2 +2x -18 = 0. However the equation directly from area=15 is x^2 +2x -3 = 15; simplified to x^2 +2x -18 = 0. The choice x^2 +2x -3 = 0 is incorrect if strictly equated to 15, but among given options the intended setup before simplification is x^2 + 2x - 3 = 15. (Follow through to x^2 +2x -18 = 0 to solve.)
24. Which quadratic has roots that are opposites of each other?
x^2 - 9 = 0 => x = ±3, which are opposites. For a quadratic with opposite roots, there is no x term (b = 0) so it looks like x^2 - k = 0.
25. If one root of 2x^2 - 3x + 1 = 0 is 1, what is the other root?
If 1 is a root, sum of roots = (-b)/a = 3/2. So other root = 3/2 - 1 = 1/2. Alternatively factor: 2x^2 -3x +1 = (2x-1)(x-1).
26. Which of these is the correct factorisation of x^2 + x - 6?
We need two numbers that multiply to -6 and add to 1: 3 and -2. So factor as (x+3)(x-2).
27. What must be true about the coefficient b in ax^2 + bx + c if the parabola is symmetric about the y-axis?
Symmetry about the y-axis means the parabola is even: f(x)=f(-x). That forces the x-term coefficient b to be 0 so the equation is ax^2 + c.