Question

The values of k for which the quadratic equation \[k x^2 + 1 = kx + 3x - 11 x^2\] has real and equal roots are

Options

• −11, −3

•  5, 7

•  5, −7

• none of these

Answer 1

5, −7
The given equation is \[k x^2 + 1 = kx + 3x - 11 x^2\] which can be written as.

\[k x^2 + 11 x^2 - kx - 3x + 1 = \]

\[ \Rightarrow \left( k + 11 \right) x^2 - \left( k + 3 \right)x + 1 = 0\]

For equal and real roots, the discriminant of

\[\left( k + 11 \right) x^2 - \left( k + 3 \right)x + 1 = 0\].

\[\therefore \left( k + 3 \right)^2 - 4\left( k + 11 \right) = 0\]

\[ \Rightarrow k^2 + 2k - 35 = 0\]

\[ \Rightarrow \left( k - 5 \right)\left( k + 7 \right) = 0\]

\[ \Rightarrow k = 5, - 7\]

Hence, the equation has real and equal roots when \[k = 5 , - 7 .\]