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.
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Solve the quadratic equation $2x^2 - 5x + 2 = 0$ by factorization.
- x = 2, x = 1/2
- x = -2, x = -1/2
- x = 2, x = -1/2
- x = -2, x = 1/2
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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 $x = 2$.
Which of the following is a quadratic equation?
- $x + 5 = 0$
- $x^2 + 2x + 1 = 0$
- $1/x + 3 = 0$
- $x^3 + x^2 + 1 = 0$
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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 ($x^3+x^2+1=0$).
If one root of $x^2 - ax + 12 = 0$ is 3, find the value of the other root.
- 4
- 9
- 3
- 12
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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.
If the roots of $3x^2 + 4x + k = 0$ are reciprocals, find k.
- 3
- 4
- 1
- 12
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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, $k = 3$.
If one root of $x^2 - 7x + k = 0$ is 4, find the value of k.
- 12
- -12
- 7
- -7
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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, $k = 12$.