SAT Question 9cb9beec: Nonlinear Equations In One Variable And Systems Of Equations In Two Variables answer and explanation with Desmos solution

According to the first equation in the given system, the value of $y$ is $-1.5$. Substituting $-1.5$ for $y$ in the second equation in the given system yields $-1.5=x^2+8x+a$. Adding $1.5$ to both sides of this equation yields $0=x^2+8x+a+1.5$. If the given system has exactly one distinct real solution, it follows that $0=x^2+8x+a+1.5$ has exactly one distinct real solution. A quadratic equation in the form $0=p{x}^2+qx+r$, where $p$, $q$, and $r$ are constants, has exactly one distinct real solution if and only if the discriminant, $q^2-4pr$, is equal to $0$. The equation $0=x^2+8x+a+1.5$ is in this form, where $p=1$, $q=8$, and $r=a+1.5$. Therefore, the discriminant of the equation $0=x^2+8x+a+1.5$ is ${\left(8\right)}^2-4\left(1\right)\left(a+1.5\right)$, or $58-4a$. Setting the discriminant equal to $0$ to solve for $a$ yields $58-4a=0$. Adding $4a$ to both sides of this equation yields $58=4a$. Dividing both sides of this equation by $4$ yields $\frac{58}{4}=a$, or $\frac{29}{2}=a$. Therefore, if the given system of equations has exactly one distinct real solution, the value of $a$ is $\frac{29}{2}$. Note that 29/2, 14.5, and 14.50 are examples of ways to enter a correct answer.