Question

Find the value of \[k\] for which the quadratic equation \[2k^2-kx+k = 0\] has equal roots.

Answer 1

Hint: Consider a quadratic equation \[ax^{2}+bx+c = 0\], where \[a\], \[b\] and \[c\] are real numbers. Its discriminant is given by \[D = b^{2}-4ac\]. Then the following cases occur:
If \[D > 0\], then the roots of the equation are real and unequal.
If \[D = 0\], then the roots of the equation are real and equal.
If \[D < 0\], then the roots of the equation are imaginary and unequal.

Complete step-by-step answer:
The quadratic equation is \[2k^2-kx+k = 0\].
Here the coefficients are \[a = 2\], \[b = -k\], and \[c = k\].
Since the given roots are real and equal, the discriminant is equal to 0.
For any quadratic equation \[ax^{2}+bx+c = 0\], its discriminant is given by \[D = b^{2}-4ac\].
This implies,
\[\begin{align*}(-k)^{2}-4(2)(k) &= 0\\ k^{2}-8k &= 0\\ k(k-8) &= 0\\ k &= 0,8\end{align*}\]
Hence the quadratic equation \[2k^2-kx+k = 0\] has equal roots for \[k = 0\] or \[k = 8\].

Note: In any quadratic equation, the coefficients are considered along with their signs. For example, in a quadratic equation \[2x^{2}-3x-2 = 0\], the coefficients are taken as \[a = 2\], \[b = -3\] and \[c = -2\].