Find the value of k, for which one root of the quadratic equation kx² - 14x + 8 = 0 is 2.

OR

Find the value(s) of k for which the equation x² + 5kx + 16 = 0 has real and equal roots.

Given: quadratic equation kx² - 14x + 8 = 0, one of its root is 2

To find: the value of ‘k’

Explanation: Given 2 is the root of the given equation hence the 2 must satisfy the given equation.

So now putting x = 2 in the given equation, we get

kx² - 14x + 8 = 0

⇒ k(2)² - 14(2) + 8 = 0

⇒ 4k - 28 + 8 = 0

⇒ 4k - 20 = 0

⇒ 4k = 20

⇒ k = 5

Hence the value of k is 5.

OR

Given: quadratic equation x² + 5kx + 16 = 0, it has real and equal roots

To find: the value of ‘k’

Explanation: The given quadratic equation has real and equal roots, so its determinant will be equal, i.e.,

D = 0

But we know determinant is b² - 4ac

Hence for real and equal roots, we get

b² - 4ac = 0………..(i)

Now comparing the given quadratic equation x² + 5kx + 16 = 0

With the standard quadratic equation ax² - bx + c = 0, we get

a = 1, b = 5k, c = 16

Substituting these values in equation (i), we get

b² - 4ac = 0

(5k)² - 4(1)(16) = 0

⇒ 25k² - 64 = 0

⇒ 25k² = 64

Taking square root on both sides, we get

Hence the value of k is