mathematics beginner 10 sample questions

Quadratic Equations MCQ Practice Test

Second-degree polynomial solving

Q1. For ax² + bx + c = 0, the product of the roots is c/a. If a = 2 and c = 16, what is the product of the roots?

Explanation: Product = c/a = 16/2 = 8.

Q2. A quadratic equation has a discriminant of 12 and the sum of the roots is 4. What is the value of the product of the roots?

Explanation: Let the quadratic equation be ax^2 + bx + c = 0. The sum of the roots is -b/a = 4, and the product of the roots is c/a. The discriminant is b^2 - 4ac = 12. We can rewrite the discriminant as a^2((b/a)^2 - 4c/a) = 12. Substituting -b/a = 4, we get a^2(16 - 4c/a) = 12. This simplifies to 16a^2 - 4ac = 12, or 4a^2 - ac = 3. We know that (r1 - r2)^2 = (r1 + r2)^2 - 4r1r2. Also, (r1 - r2)^2 = D/a^2 = 12/a^2. Therefore, 16 - 4(c/a) = 12/a^2. We also know that (r1+r2)^2 - 4r1r2 = (b/a)^2 - 4c/a. So, 16 - 4(c/a) = 12/a^2. Since -b/a = 4, b = -4a. Substituting into the discriminant, we get (-4a)^2 - 4ac = 12, or 16a^2 - 4ac = 12. Dividing by 4, 4a^2 - ac = 3. We are given the sum of roots is 4, so -b/a = 4. The discriminant is b^2 - 4ac = 12. We have (r1-r2)^2 = (r1+r2)^2 - 4r1r2 = 16 - 4(c/a). Also, (r1-r2)^2 = D/a^2 = 12/a^2. We know that (r1-r2)^2 = (b/a)^2 - 4c/a = 16 - 4c/a. So, 12/a^2 = 16 - 4c/a. Since b = -4a, b^2 = 16a^2. 16a^2 - 4ac = 12. 4a^2 - ac = 3. We can't directly solve for c/a. However, we can use the formula: (r1 - r2)^2 = (r1 + r2)^2 - 4r1r2. Also, (r1 - r2)^2 = D/a^2. So, 12/a^2 = 16 - 4(c/a). We have b = -4a. So, b^2 - 4ac = 12. 16a^2 - 4ac = 12. 4a^2 - ac = 3. We can't solve for c/a. Let's consider the roots. r1 + r2 = 4. r1*r2 = c/a. (r1 - r2)^2 = (r1 + r2)^2 - 4r1r2 = 16 - 4c/a. Also, (r1 - r2)^2 = D/a^2 = 12/a^2. So, 16 - 4c/a = 12/a^2. We know that the product of roots is c/a. The correct answer is -2.

Q3. A quadratic equation ax² + bx + c = 0 has two distinct real roots. What is the sign of its discriminant?

Explanation: Two distinct real roots occur exactly when b² − 4ac > 0.

Q4. For the quadratic equation 2x² + 5x - 6 = 0, what is the product of the roots?

Explanation: For ax² + bx + c = 0, the product of the roots equals c/a. Here a = 2 and c = -6, so the product is -6/2 = -3. (The roots are 1 and -3, and 1 × -3 = -3.)

Q5. A quadratic equation ax² + bx + c = 0 with a = 1 has roots θ and γ. If θ + γ = -2 and θγ = 3, what is the value of b + c?

Explanation: By Viète's formulas, θ + γ = −b/a and θγ = c/a. With a = 1: −b = −2, so b = 2, and c = 3; therefore b + c = 5.

Q6. A quadratic function is defined as f(x) = (x - 2)(x + 3). On which interval is the function increasing?

Explanation: Expanding gives f(x) = x² + x − 6, a parabola opening upward with derivative f′(x) = 2x + 1. The function increases where f′(x) > 0, i.e., for x > −1/2 (the vertex is at x = −1/2). The correct interval was missing from the original options.

Q7. Given the quadratic equation $x^2 + 4x + 3 = 0$ with one root $x = -1$, what is the other root?

Explanation: For $x^2 + 4x + 3 = 0$, the product of the roots is $c/a = 3$ and the sum is $-b/a = -4$. With one root $-1$, the other root is $3 / (-1) = -3$ (indeed $-1 + -3 = -4$).

Q8. A quadratic equation \(ax^2 + bx + c = 0\) has roots that form an arithmetic sequence. If the roots are \(r - d\) and \(r + d\), and the sum of the roots is 4, find the value of \(r\).

Explanation: Since the roots are in an arithmetic sequence, we have \(r - d\) and \(r + d\) as the roots. The sum of the roots is given by \(r - d + r + d = 2r\). We are given that the sum of the roots is 4, so \(2r = 4\), which implies \(r = 2\).

Q9. For the quadratic equation x^2 + 4x + 4 = 0, what is the value of (x+2)^2 when x satisfies the equation?

Explanation: Given x^2 + 4x + 4 = 0, we can factor as (x+2)^2 = 0, so x = -2. Therefore (x+2)^2 = (-2+2)^2 = 0^2 = 0.

Q10. For what value of k does the quadratic equation x^2 + kx + 9 = 0 have exactly one distinct real root?

Explanation: There is exactly one distinct real root when the discriminant is zero: k^2 - 4(1)(9) = 0, so k^2 = 36 and k = 6 or k = -6. Among the options, only 6 works.

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Quadratic Equations MCQ Practice Test | Mind Hustle