Final test!

$\large \displaystyle \text{The Equation is }4x^2 + 6x + 2 = 0.\\ \large \displaystyle \text{We can find the roots using the formula } x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\ \large \displaystyle \text{Where a,b,c are the numbers in the equation, where the equation is expressed as }ax^2 + bx + c = 0.\\ \large \displaystyle x = \frac{-6 \pm \sqrt{4}}{8} = -\frac{1}{2} and -1.\\ \large \displaystyle \text{Where u is the largest root and z is the smallest root}\\ \large \displaystyle \therefore u = -\frac{1}{2} and z = -1.\\ \large \displaystyle \text{By substituting the value of u and z in the given equation}\\ \large \displaystyle \text{We get } \frac{-57k + 1 + 513k - 9}{3 \times 8} = 18 + \frac{2}{3}\\ \large \displaystyle \implies 456k - 8 = 24 \times (18 + \frac{2}{3})\\ \large \displaystyle \implies 456k - 8 = 432 + 16\\ \large \displaystyle \implies 456k = 456\\ \large \displaystyle \implies k = \color{#D61F06}{\boxed1}.$

Greeting for @Hung Woei Neoh , for correcting my error. $\ddot\smile$

Given 4x^2+6x+2 = 0. by putting 4,2,6 in the quadratic equation we find that the roots are (-0.5,-1). after finding the roots, we can see that u = (-0.5), and z = (-1). after that, we simplfy (114ul +2uz +513k +9z) to (2u+9)(57k+z) by factoring. we cancel (2u +9) by dividing both denominator and numerator by (2u +9), we are left with (57k-1)/(3) = 18+2/3. by solving the linear equation we find k = 1. we can also check by putting 1 in the original equation.

Thanks for trying my problem!