If alpha and beta are the zeroes of the quadratic polynomial p(x)=x^2-(k+ 6)+2(2k-1),then find the value of k if alpha plus beta is equal to alpha ×beta by 2?

Question

Answer

Answer

Explanation:

Let p(x) be a quadratic polynomial, then the general form of the quadratic polynomial is:

p(x) = ax^2 + bx + c

where a, b, and c are the coefficients of the polynomial.

Now, let alpha and beta be the roots of the given quadratic polynomial, then we have:

p(alpha) = 0 and p(beta) = 0

Using the factor theorem, we can write:

p(x) = (x - alpha)(x - beta)

Expanding the above expression, we get:

p(x) = x^2 - (alpha + beta)x + alpha * beta

Comparing the coefficients of the above equation with the given quadratic polynomial, we get:

a = 1, b = -(alpha + beta), and c = alpha * beta

Solution:

Given quadratic polynomial is:

p(x) = x^2 - (k * 6) * 2(2k - 1)

On comparing the coefficients, we get:

a = 1, b = -(alpha + beta) = (k * 6) * 2(2k - 1), and c = alpha * beta = 0.5(alpha + beta)

As alpha + beta = 2c, we get:

alpha + beta = 2 * 0.5(alpha + beta)

alpha + beta = alpha * beta

Substituting the value of b, we get:

(k * 6) * 2(2k - 1) = -(alpha + beta)

(k * 6) * 2(2k - 1) = -alpha - beta

Solving the above equation, we get:

k = 0 or k = 5/6

Therefore, the value of k is either 0 or 5/6.

Answer

Alpha×beta / 2 =alpha+beta
k+6=2k-1
k=7

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