SAT Question 6ce95fc8: Nonlinear Equations In One Variable And Systems Of Equations In Two Variables answer and explanation with Desmos solution

Choice D is correct. A quadratic equation in the form $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants, can be solved using the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Subtracting $2x+3$ from both sides of the given equation yields $2x^2-2x-5=0$. Applying the quadratic formula, where $a=2$, $b=-2$, and $c=-5$, yields $x=\frac{-(-2)\pm \sqrt{(-2{)}^2-4\times 2\times (-5)}}{2\times 2}$. This can be rewritten as $x=\frac{2\pm \sqrt{44}}{4}$. Since $\sqrt{44}=\sqrt{2^2\times 11}$, or $2\times \sqrt{11}$, the equation can be rewritten as $x=\frac{2\pm 2\times \sqrt{11}}{4}$. Dividing $2$ from both the numerator and denominator yields $\frac{1+\sqrt{11}}{2}$ or $\frac{1-\sqrt{11}}{2}$. Of these two solutions, only $\frac{1+\sqrt{11}}{2}$ is present among the choices. Thus, the correct choice is D.