SAT Question 5da5c665: Nonlinear Equations In One Variable And Systems Of Equations In Two Variables answer and explanation with Desmos solution

Choice B is correct. The solutions to a quadratic equation of the form $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants, can be calculated using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. The given equation is of this form, where $a=4$, $b=-5$, and $c=-5$. Substituting these values into the quadratic formula yields $x=\frac{-\left(-5\right)\pm \sqrt{(-5{)}^2-4\left(4\right)\left(-5\right)}}{2(4)}$, or $x=\frac{5\pm \sqrt{25+80}}{8}$. These solutions can be rewritten as $x=\frac{5\pm \sqrt{105}}{8}$, or $x=\frac{5}{8}\pm \frac{\sqrt{105}}{8}$. Since $\frac{5}{8}+\frac{\sqrt{105}}{8}$ is greater than $\frac{5}{8}-\frac{\sqrt{105}}{8}$, it follows that the greatest solution to the given equation is $\frac{5}{8}+\frac{\sqrt{105}}{8}$.