Question

Find the values of k for which the following equations have real and equal roots:
(i) $x^2-2\left(k+1\right)x+k^2=0$
(ii) $k^2{x}^2-2\left(2k-1\right)x+4=0$
(iii) $\left(k+1\right)x^2-2\left(k-1\right)x+1=0$
(iv) $x^2+k\left(2x+k-1\right)+2=0$

Answer 1

(i) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

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

Therefore, the value of

(ii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

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

Therefore, the value of

(iii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

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

Now factorizing of the above equation

So, either

Therefore, the value of

(iv) The given equation is $x^2+k\left(2x+k-1\right)+2=0$.
$\Rightarrow x^2+2kx+k\left(k-1\right)+2=0$
So, a = 1, b = 2k, c = k(k − 1) + 2
We know $D=b^2-4ac$
$$\begin{gathered} \Rightarrow D={\left(2k\right)}^2-4\times 1\times \left[k\left(k-1\right)+2\right] \\ \Rightarrow D=4k^2-4\left[k^2-k+2\right] \\ \Rightarrow D=4k^2-4k^2+4k-8 \\ \Rightarrow D=4k-8=4\left(k-2\right) \end{gathered}$$
For equal roots, D = 0
Thus, 4(k − 2) = 0
So, k = 2.