Quadratic Equations MCQ Test: Quiz!

Explanation
The equation will have equal roots when the discriminant is equal to zero. The discriminant is given by b^2 - 4ac. In this case, a = 1, b = 0, and c = -k. Substituting these values into the discriminant formula, we get 0 - 4(1)(-k) = 4k. For the equation to have equal roots, 4k must equal zero, which means k must equal zero. Therefore, the correct answer is ±3.

Explanation
Since the equation has two roots, x = -1 and x = -2, we can substitute these values into the equation to find the values of p and q.

When x = -1, we have p(-1)^2 + 3(-1) + q = 0. Simplifying this equation gives p - 3 + q = 0.

Similarly, when x = -2, we have p(-2)^2 + 3(-2) + q = 0. Simplifying this equation gives 4p - 6 + q = 0.

Now we can solve these two equations simultaneously. Subtracting the first equation from the second equation eliminates q and gives 4p - p - 6 + 3 = 0, which simplifies to 3p - 3 = 0. Solving for p gives p = 1.

Substituting p = 1 into the first equation, we have 1 - 3 + q = 0. Simplifying this equation gives q = 2.

Therefore, the value of q - q is 2 - 2 = 0.

Explanation
The given options are potential solutions to a quadratic equation. Without the equation itself, it is not possible to determine which of the options is the correct solution. Therefore, an explanation cannot be provided.

Explanation
The given question asks for the value of k that makes x = -2 a root of the quadratic equation. In a quadratic equation, the roots are the values of x that make the equation equal to zero. Therefore, if x = -2 is a root, substituting -2 into the equation should result in zero. By substituting -2 into the equation and simplifying, we find that the value of k that satisfies this condition is 2.

Explanation
The correct answer is 5,-2 because the equation a - 3 = 0 can be rearranged to a = 3. Therefore, the value of a is 3. Additionally, the equation a - 3 = 5 can be rearranged to a = 8. Therefore, the value of a is also 8. However, the equation a - 3 = -2 can be rearranged to a = 1. Therefore, the value of a is not -2.

Explanation
To find the value of p so that the equation has no real root, we need to consider the discriminant of the quadratic equation. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. In this case, the equation is x^2 - px + 12 = 0. Comparing it with the standard form ax^2 + bx + c = 0, we can see that a = 1, b = -p, and c = 12. For the equation to have no real root, the discriminant must be negative. Therefore, we have (-p)^2 - 4(1)(12) < 0. Simplifying this inequality, we get p > 8.

Explanation
The given question is incomplete and does not provide any information about the quadratic equation or the value of k. Therefore, it is not possible to generate an explanation for the given correct answer.

Explanation
The common root of the quadratic equations x^2 - 3x + 2 = 0 and 2x^2 - 5x + 2 = 0 is x = 2. This can be found by solving both equations separately and finding the value of x that satisfies both equations. By substituting x = 2 into both equations, we get 2^2 - 3(2) + 2 = 0 and 2(2)^2 - 5(2) + 2 = 0, which simplifies to 0 = 0 in both cases. Therefore, x = 2 is the common root.

Explanation
The difference of the roots of a quadratic equation is equal to the negative coefficient of the linear term divided by the coefficient of the quadratic term. In this case, the difference of the roots is given as 1. Therefore, we can set up the equation k/1 = -1, which simplifies to k = -1. Since we are looking for the positive value of k, the correct answer is 7.

Explanation
If the roots of a quadratic equation are equal, it means that the discriminant (b^2 - 4ac) is equal to zero. In this case, the equation is kx(x - 2) + 6 = 0. By expanding and rearranging the equation, we get kx^2 - 2kx + 6 = 0. Comparing this with the standard form of a quadratic equation ax^2 + bx + c = 0, we can see that a = k, b = -2k, and c = 6. To find the discriminant, we substitute these values into the formula and set it equal to zero: (-2k)^2 - 4(k)(6) = 0. Simplifying this equation, we get 4k^2 - 24k = 0. Factoring out k, we get k(4k - 24) = 0. Therefore, k = 0 or k = 6. Since k cannot be zero (as it would make the equation linear), the value of k for the quadratic equation with equal roots is 6.

Explanation
For the roots of an equation to be equal, the discriminant (b^2 - 4ac) must be equal to zero. In this equation, a = 2, b = -k, and c = 1. Plugging these values into the discriminant formula, we get (-k)^2 - 4(2)(1) = k^2 - 8. To have equal roots, k^2 - 8 must be equal to zero. Solving for k, we find that k = ±√8 or k = ±2√2.

Explanation
The equation has two distinct real roots because it is a quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. In this case, the equation has two distinct real roots because the discriminant (b^2 - 4ac) is positive, indicating that there are two different solutions for x.