Question

Find the value of k for which the roots of the equation 8kx ( x - 1 ) + 1 = 0 are real and equal.

Answer 1

b^2 =-4ac is equal to zero then the roots are real and equal
and if it is less than zero then it is real and unequal

Answer 2

If a Quadratic Equation ax² + bx + c = 0 has Real and Equal Roots then the Discriminant of the respective Equation is Zero.

⇒ Discriminant : b² - 4ac = 0

The Given Quadratic Equation is 8kx(x - 1) + 1 = 0

⇒ 8kx² - 8kx + 1 = 0

Comparing with the Standard form we can notice that :

a = 8k and b = -8k and c = 1

Substituting the above Values in b² - 4ac = 0

⇒ b² - 4ac = 0

⇒ (-8k)² - 4(8k)(1) = 0

⇒ 64k² - 32k = 0

⇒ 64k² = 32k

⇒ 2k = 1

⇒ k = 1/2

Verification :

substituting the Value of k in the given Quadratic Equation we get :

⇒ 8kx(x - 1) + 1 = 0

⇒ 8(1/2)x(x - 1) + 1 = 0

⇒ 4x(x - 1) + 1 = 0

⇒ 4x² - 4x + 1 = 0

⇒ (2x - 1)² = 0

⇒ 2x - 1 = 0

⇒ x = 1/2

⇒ The Roots of Quadratic Equation are 1/2 , 1/2

The Given Quadratic Equation has Equal roots for the Value of k = 1/2

⇒ k = 1/2 is True