Question

Q5. (1) One root of the equation 4x ^ 2 12x k = 0 k \in \mathbb{T} and x \in \mathbb{R} is five times the other root. Find the value of k.

Answer 1

The value of k for the quadratic equation 4x^2 + 12x + k = 0, where one root is five times the other, is found to be 5.

Step-by-step explanation:

The given quadratic equation is
`4x^2 + 12x`+ k = 0. We are told that one root is five times the other. Let's assume the roots are x and 5x. By the relationship of roots and coefficients for quadratic equations (which states that the sum of the roots is equal to -b/a, and the product of the roots is c/a), we can set up two equations.

The sum of the roots will be x + 5x = 6x, which is equal to -12/4 = -3 (simplifying -b/a). Therefore, we have 6x = -3 which simplifies to x = -1/2.

The product of the roots will be
`x * 5x = 5x^2,`which is equal to k/4 (simplifying c/a). Substituting x = -1/2 we get
`5 * (-1/2)^2 = k/4,`this simplifies to 5/4 = k/4, which leads us to the value of k = 5.

Therefore, the value of k for which one root of the equation is five times the other is k = 5.