Question

Find the values of k for which 2x^2-3x+k^2-3k+4 attains all Positive and negative values
I WILL MARK BRAINLIEST

Answer 1

$we \: know \: for \: quadratic \: equation \: - \\ \\ = d \geqslant 0 \: \: \: \: : \bold \green{ \: for \: real \: values} \\ \\ = \sqrt{ {b}^{2} - 4ac} \geqslant 0 \\ \\ = \sqrt{ {3}^{2} - 4 \times 2 \times ( {k}^{2} - 3k + 4)} \geqslant 0 \\ \\ = \sqrt{9 - 8 {k}^{2} + 24k - 32} \geqslant 0 \\ \\ = - 8 {k}^{2} + 24k - 23 \geqslant 0 \\ \\ = 8 {k}^{2} - 24k + 23 \leqslant 0 \\ \\ 8 {k}^{2} - 24k + 23 = 0 \\ \\ k = \frac{ 24 + \sqrt{576 - 4 \times 23 \times 8} }{2 \times 8} \\ \\ k = \frac{24 + \sqrt{24 \times 24 - 4 \times 8 \times 23} }{2 \times 8} \\ \\ k = \frac{4(6 + \sqrt{36 - 46)} }{2 \times 8} \\ \\ \bold \red{here \: we \: find \: the \: values \: are \: not \: real} \\ \\ \bold \green{hence \: there \: is \: no \: values \: of \: k \: which \: is \: real}$