Question

Find the integral values of $k$ for which the equation $(k-12)x^2+2(k-12)x+2=0$ possesses no real roots.

Answer 1

A quadratic equation has not real roots if $D<0$

$\Rightarrow b^2-4ac<0$

$\Rightarrow$ $\left(2\right(k-12){)}^2-4\times (k-12)\times 2<0$

$\Rightarrow$ $4(k^2+144-24k)-8k+96<0$

$\Rightarrow$ $4k^2+576-96k-8k+96<0$

$\Rightarrow$ $4k^2-104k+672<0$

$\Rightarrow k^2-26k+168<0$

$\Rightarrow (k-12)(k-14)<0$

$\Rightarrow k-12<0$ and $k-14>0$ or $k-12>0$ and $k-14<0$

$\Rightarrow k>12$ and $k<14$

$\Rightarrow 12<k<14$

Hence, $k\in (12,14)$.