Quadratic Equations Quiz - ICSE Class 8 Mathematics

Free practice quiz: Quadratic Equations (ICSE Class 8 Mathematics). 52 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 8 Mathematics practice quiz has 52 multiple-choice questions on Quadratic Equations, and 52 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. Solve the quadratic equation $2x^2 - 5x + 2 = 0$ by factorization.

   1. x = 2, x = 1/2
   2. x = -2, x = -1/2
   3. x = 2, x = -1/2
   4. x = -2, x = 1/2
   Show answer

   Answer: A. x = 2, x = 1/2

   We need to find two numbers that multiply to $2 imes 2 = 4$ and add up to -5. These numbers are -4 and -1. So, we can rewrite the middle term: $2x^2 - 4x - x + 2 = 0$. Now, factor by grouping: $2x(x - 2) - 1(x - 2) = 0$. This gives $(2x - 1)(x - 2) = 0$. Setting each factor to zero, we get $2x - 1 = 0$ or $x - 2 = 0$. Solving these, we get $x = 1/2$ or $x = 2$.

2. Which of the following is a quadratic equation?

   1. $x + 5 = 0$
   2. $x^2 + 2x + 1 = 0$
   3. $1/x + 3 = 0$
   4. $x^3 + x^2 + 1 = 0$
   Show answer

   Answer: B. $x^2 + 2x + 1 = 0$

   A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (x in this case) is 2. The equation $x^2 + 2x + 1 = 0$ has the highest power of x as 2, making it a quadratic equation. The other options are linear ($x+5=0$), rational ($1/x+3=0$), or cubic ($x^3+x^2+1=0$).

3. If one root of $x^2 - ax + 12 = 0$ is 3, find the value of the other root.

   1. 4
   2. 9
   3. 3
   4. 12
   Show answer

   Answer: A. 4

   Let the roots be $\alpha$ and $eta$. We are given $\alpha = 3$. For the quadratic equation $x^2 - ax + 12 = 0$, the product of the roots is $\alphaeta = c/a = 12/1 = 12$. Substituting $\alpha = 3$, we get $3eta = 12$. Solving for $eta$, we find $eta = 12/3 = 4$. The other root is 4. We can also find 'a'. The sum of the roots is $\alpha + eta = -(-a)/1 = a$. So, $3 + 4 = a$, which means $a = 7$. The equation is $x^2 - 7x + 12 = 0$, and its roots are indeed 3 and 4.

4. If the roots of $3x^2 + 4x + k = 0$ are reciprocals, find k.

   1. 3
   2. 4
   3. 1
   4. 12
   Show answer

   Answer: A. 3

   For the roots of a quadratic equation $ax^2 + bx + c = 0$ to be reciprocal, the product of the roots must be 1. The product of the roots is given by $c/a$. In this equation, a = 3, b = 4, and c = k. So, for the roots to be reciprocal, $c/a = 1$, which means $k/3 = 1$. Therefore, $k = 3$.

5. If one root of $x^2 - 7x + k = 0$ is 4, find the value of k.

   1. 12
   2. -12
   3. 7
   4. -7
   Show answer

   Answer: A. 12

   Since x = 4 is a root, substitute it into the equation: $(4)^2 - 7(4) + k = 0$. This simplifies to $16 - 28 + k = 0$. Combining constants, we get $-12 + k = 0$. Therefore, $k = 12$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. The sum of the roots of the quadratic equation $3x^2 + 6x + 5 = 0$ is:
2. If the product of the roots of the quadratic equation $x^2 - 7x + k = 0$ is 12, find the value of k.
3. Which of the following quadratic equations has roots 3 and -5?
4. Find the roots of the quadratic equation $x^2 - 9 = 0$ using the quadratic formula.
5. If one root of the quadratic equation $2x^2 + ax + 6 = 0$ is 2, find the value of 'a'.
6. The difference between the roots of the quadratic equation $x^2 - 5x + 6 = 0$ is:
7. If the roots of the quadratic equation $ax^2 + bx + c = 0$ are real and distinct, then:
8. Find the value of 'k' for which the equation $x^2 - kx + 4 = 0$ has equal roots.
9. If one root of the quadratic equation $x^2 - 5x + m = 0$ is 2, find the other root.
10. If the roots of the equation $3x^2 - 8x + 5 = 0$ are $\alpha$ and $eta$, then the value of $\alphaeta$ is:
11. Solve the equation $(x+2)(x+3) = 0$.
12. If the discriminant of a quadratic equation is 0, the roots are:
13. Find the sum of the roots of the quadratic equation $2x^2 + 7x + 3 = 0$.
14. If $x^2 + 5x + 6 = 0$, then the values of x are:
15. The product of the roots of $x^2 - 3x + 2 = 0$ is:
16. What are the roots of the quadratic equation $x^2 - 4x + 4 = 0$?
17. If the roots of the quadratic equation $x^2 + bx + c = 0$ are 4 and 5, what is the value of c?
18. Find the value of k for which the quadratic equation $kx^2 - 5x + k = 0$ has equal roots.
19. For what value of m will the roots of $4x^2 + mx + 9 = 0$ be equal?
20. If $\alpha$ and $eta$ are the roots of $x^2 + 6x + 8 = 0$, find $\alpha + eta$.
21. Solve the quadratic equation $3x^2 - 12x = 0$ by factorization.
22. If the discriminant of $ax^2 + bx + c = 0$ is negative, the roots are:
23. Find the value of k such that $x^2 - 6x + k = 0$ has equal roots.
24. If the roots of the quadratic equation $x^2 - 8x + p = 0$ are equal, find the value of p.
25. Which quadratic equation has roots $\sqrt{3}$ and $-\sqrt{3}$?
26. Find the roots of $2x^2 - 3x - 2 = 0$ by completing the square.
27. If the sum of the roots of $kx^2 + 2x + 3 = 0$ is -1, find the value of k.
28. For which value of p does the equation $x^2 - px + 4 = 0$ have real and distinct roots?
29. Solve $x^2 - x - 20 = 0$ using the quadratic formula.
30. If the roots of $x^2 - (k+1)x + 5 = 0$ are $\alpha$ and $eta$, and $\alpha + eta = 4$, find k.
31. What is the nature of the roots of the equation $x^2 + 2x + 5 = 0$?
32. Find the roots of the quadratic equation $x^2 - 7x + 10 = 0$ by factorization.
33. If the sum of the roots of $2x^2 + bx + 5 = 0$ is 3, find the value of b.
34. Which of the following equations has roots 2 and -3?
35. Find the value of k if the equation $3x^2 - 18x + k = 0$ has equal roots.
36. If the product of the roots of $x^2 + 5x + k = 0$ is 10, find the value of k.
37. Solve the quadratic equation $4x^2 - 12x + 9 = 0$ by factorization.
38. Find the value of k for which the equation $x^2 + kx + 9 = 0$ has no real roots.
39. If $\alpha$ and $eta$ are the roots of $x^2 - 5x + 6 = 0$, find the value of $\alphaeta$.
40. Solve the quadratic equation $x^2 - 2x - 8 = 0$ by completing the square.
41. Find the sum of the squares of the roots of $x^2 - 5x + 6 = 0$.
42. If the equation $x^2 + 2x + k = 0$ has real and distinct roots, then:
43. What is the value of k if the roots of $2x^2 + 5x + k = 0$ are equal?
44. If one root of $x^2 + kx - 15 = 0$ is 3, find the other root.
45. Find the value of k if the roots of $x^2 - 3x + k = 0$ are such that one root is twice the other.
46. Find the roots of $x^2 - 6x + 5 = 0$ using the quadratic formula.
47. If one root of $x^2 - ax + 8 = 0$ is 2, find the value of a.

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