Category: Algebra MCQs

Correct Answer: C. 28x^2 + 49x -16 = 0

If the roots of the equation are -7/4 and 4/7, then the equation can be written as:

(x + 7/4)(x – 4/7) = 0

Multiplying out the terms gives:

x^2 + (7/4)x – (4/7)x – 7/4 * 4/7 = 0
x^2 + (49x – 16)/28 = 0

Multiplying both sides by 28 to get rid of the denominator, we get:

28x^2 + 49x – 16 = 0

Therefore, the equation with roots -7/4 and 4/7 is 28x^2 + 49x - 16 = 0.

Correct Answer: D. x^2-Sx+P=0

If we have a quadratic equation of the form ax^2 + bx + c = 0, the sum of its roots is given by:

S = -b/a

and the product of its roots is given by:

Correct Answer: C. Irrational

If the discriminant of the quadratic equation ax^2 + bx + c = 0 is not a perfect square, then its roots are irrational and real.

The discriminant of the quadratic equation is given by the expression b^2 - 4ac. If this value is not a perfect square, then the roots of the equation will be of the form:

(-b ± √(b^2 – 4ac)) / 2a

Since the discriminant is not a perfect square, the square root of b^2 - 4ac will be irrational, and the roots will be of the form (-b ± irrational number) / 2a, which are both real and irrational. Therefore, the answer is option C: Irrational.

Correct Answer: D. 7/122

The given quadratic equation is:

122x^2 – 7x + 4 = 0

We can find the sum of the roots of this equation using the formula:

sum of roots = -b/a

where a and b are the coefficients of x^2 and x, respectively.

Comparing the given equation with the standard form ax^2 + bx + c = 0, we can see that a = 122 and b = -7.

So, the sum of the roots of the equation 122x^2 – 7x + 4 = 0 is:

sum of roots = -b/a = -(-7) / 122 = 7/122

Therefore, the sum of the roots of the equation 122x^2 – 7x + 4 = 0 is 7/122.

Correct Answer: C. Real & unequal

If the discriminant of a quadratic equation ax^2 + bx + c = 0 is positive (i.e., b^2 – 4ac > 0), then the roots of the quadratic equation are real and unequal.

This is because the discriminant is used to determine the nature of the roots of a quadratic equation. Specifically, if the discriminant is positive, then the roots of the quadratic equation are real and unequal. This is because in this case, the quadratic formula (-b ± sqrt(b^2 – 4ac)) / 2a will yield two distinct real solutions for x.

In contrast, if the discriminant is zero (i.e., b^2 – 4ac = 0), then the roots of the quadratic equation are real and equal. This is because the quadratic formula will yield a single real solution for x.

If the discriminant is negative (i.e., b^2 – 4ac < 0), then the roots of the quadratic equation are complex conjugates. This means that the roots will have the form of a+bi and a-bi, where a and b are real numbers and i is the imaginary unit.

Correct Answer: A. -b/a

The sum of the roots of a quadratic equation of the form ax^2 + bx + c = 0 is given by -b/a.

To see why this is the case, let the roots of the quadratic be x1 and x2. Then we can write the quadratic equation as:

ax^2 + bx + c = a(x – x1)(x – x2) = 0

Expanding the right-hand side gives:

a(x^2 – (x1 + x2)x + x1x2) = 0

Comparing the coefficients of x^2, x, and the constant term on both sides, we get:

x^2 coefficient: a = a x coefficient: -a(x1 + x2) = b constant term: a(x1)(x2) = c

Solving for x1 + x2, we have:

x1 + x2 = -b/a

Therefore, the sum of the roots of the quadratic equation ax^2 + bx + c = 0 is -b/a.

Correct Answer: D. -3

To find the product of the roots of the quadratic equation 9x^2-5x-27=0, we can use the fact that the product of the roots of a quadratic equation ax^2+bx+c=0 is equal to c/a.

Therefore, in the given equation 9x^2-5x-27=0, we can see that a=9, b=-5, and c=-27.

So, the product of the roots will be:

c/a = (-27) / (9) = -3

Hence, the product of the roots of the equation 9x^2-5x-27=0 is -3.

Correct Answer: D. 11

If y^4 + y^3 – 2y^2 – 8y + 2 is divided by y + 1 using synthetic division, we get:

-1 | 1 1 -2 -8 2
| -1 3 -1 9
+————–
| 1 0 -2 -9 11

The remainder is the last number in the last row, which is 11. Therefore, the correct option is D. 11

Correct Answer: A. 2

A quadratic equation is an equation of degree 2. The term “quadratic” comes from the Latin word “quadratus,” which means “square.” A quadratic equation has the form ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to 0. The highest power of the variable x in a quadratic equation is 2, which makes it a degree 2 polynomial equation. Therefore, the answer is A. 2.

Correct Answer: B. Trinomial

The expression 3x^2 + 9x + 7 is a trinomial, which is a polynomial with three terms. In this case, the three terms are 3x^2, 9x, and 7.