The possible values of k are k < - 1.868 or k > 0.535 in which the inequality is true.
What is a quadratic equation?
Any equation of the form [tex]\rm ax^2+bx+c=0[/tex] where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.
As we know, the formula for the roots of the quadratic equation is given by:
[tex]\rm x = \dfrac{-b \pm\sqrt{b^2-4ac}}{2a}[/tex]
We have a quadratic equation:
(k - 1)x² + (4k)x + k -3 = 0
Here,
a = k - 1
b = 4k
c = k - 3
As we know for distinct real roots:
D > 0
[tex]\rm {b^2-4ac}} > 0[/tex]
(4k)² - 4(k-1)(k-3) > 0
[tex]\rm 16k^2-4\left(k-1\right)\left(k-3\right) > 0[/tex]
[tex]\rm 12k^2+16k-12 > 0[/tex]
[tex]\rm 3k^2+4k-3 > 0[/tex]
[tex]\rm 3\left(k+\dfrac{2}{3}\right)^2-\dfrac{13}{3} > 0[/tex]
[tex]\rm 3\left(k+\dfrac{2}{3}\right)^2 > \dfrac{13}{3}[/tex]
[tex]\rm \left(k+\dfrac{2}{3}\right)^2 > \dfrac{13}{9}[/tex]
[tex]\rm k < \dfrac{-\sqrt{13}-2}{3}\quad \mathrm{or}\quad \:k > \dfrac{\sqrt{13}-2}{3}[/tex]
or
[tex]\rm k < -1.868 \quad \mathrm{or}\quad \:k > 0.535[/tex]
Thus, the possible values of k are k < - 1.868 or k > 0.535 in which the inequality is true.
Learn more about quadratic equations here:
brainly.com/question/2263981
#SPJ1
