#### Question

Find that non-zero value of *k*, for which the quadratic equation *kx*^{2} + 1 − 2(*k* − 1)*x* + *x*^{2} = 0 has equal roots. Hence find the roots of the equation.

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#### Solution

We have

kx^{2} +1−2(k−1)x+x^{2}=0

This equation can be rearranged as

(k+1)x^{2} −2(k−1)x+1=0

Here, *a *=* k *+ 1*, b = *−2(*k *− 1) and *c *= 1

∴ D = *b*^{2} − 4*ac*

=[−2(k−1)^{2}]−4×(k+1)×1

=4(k−1)^{2}−4(k+1)

=4[(k−1)^{2} −k−1]

=4[k^{2} +1−2k−k−1]

=4[k^{2}−3k]

=4[k(k−3)]

The given equation will have equal roots, if D = 0

⇒ 4[k(k−3)] = 0

⇒ *k* = 0 or *k *− 3 = 0

⇒* k = *3

Putting *k = *3 in the given equation, we get

4x^{2}−4x+1=0

⇒(2x−1)^{2}=0

⇒2x−1=0

`=>x=1/2`

Hence, the roots of the given equation are `1/2 " and "1/2`

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#### Reference Material

Solution for question: Find that non-zero value of k, for which the quadratic equation kx^2 + 1 − 2(k − 1)x + x^2 = 0 has equal roots. Hence find the roots of the equation. concept: Nature of Roots. For the course CBSE