NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 2 (ii)

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 2 (ii)

Question:

Find the values of k for each of the following quadratic equations, so that they have two equal roots : kx (x – 2) + 6 = 0.

Given:

The equation: kx (x – 2) + 6 = 0

To Find:

The value(s) of 'k' such that the equation has two equal roots.

Formula:

A quadratic equation of the form ax2 + bx + c = 0 to have two equal roots, its discriminant (Δ) must be equal to zero.

Δ = b2 - 4ac = 0

Solution:

The given equation is kx (x – 2) + 6 = 0.
First, rewrite the equation in the standard quadratic form ax2 + bx + c = 0:
⇒ kx2 - 2kx + 6 = 0
Comparing it with the standard form ax2 + bx + c = 0, we have:

a = k

b = -2k

c = 6
For the equation to be a quadratic equation, the coefficient 'a' cannot be zero, i.e., k ≠ 0. (Condition 1)
For two equal roots, the discriminant must be zero (Δ = 0):

⇒ b2 - 4ac = 0

⇒ (-2k)2 - 4(k)(6) = 0

⇒ 4k2 - 24k = 0

Factor out 4k from the equation:

⇒ 4k(k - 6) = 0

This gives two possible solutions for k:

4k = 0 or k - 6 = 0

⇒ k = 0 or k = 6
As established, for the given equation to be a quadratic equation, k cannot be 0 as per (Condition 1)

Therefore, we discard k = 0 so k = 6.

Result:

For k = 6, the quadratic equation kx (x – 2) + 6 = 0 has two equal roots.
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