Question

Given the quadratic equation (4(x-2)^2=k(x-1)), show that the sum of the roots is equal to their product. Find the values of the constant (k) for which the difference between the roots is (3 times (3)(4)).

a) (k=3)
b) (k=12)
c) (k=6)
d) (k=9)

Answer 1

Expanding the given quadratic equation allows us to find the standard form ax^2 + bx + c = 0. The sum and product of the roots are both (16+k)/4. To find k for which the root difference is 36, we use the relationship between the root difference and the discriminant.

The correct answer is D.

Step-by-step explanation:

To show that the sum of the roots of the given quadratic equation (4(x-2)^2 = k(x-1)) is equal to their product, we need to first expand and put the equation in standard form (ax^2 + bx + c = 0). After doing so, the sum and product of the roots can be determined by the formulas -b/a and c/a, respectively, which are derived from the quadratic formula.

Expanding the given equation, we get:

4(x^2 - 4x + 4) = kx - k

4x^2 - 16x + 16 = kx - k

4x^2 - (16+k)x + (16+k) = 0

The sum of the roots (α + β) would be equal to (16+k)/4 and the product (αβ) would also be (16+k)/4 since the constant term (c) is (16+k).

To find the values of k for which the difference between the roots is 3 times (3)(4), we need to set the difference of the roots (α - β) to 36 (since 3 times 12 equals 36). This involves the relationship between the difference of the roots and the discriminant (√(b^2 - 4ac)). By solving for k, we can determine the correct answer.