SAT Question f880f910: Nonlinear Functions answer and explanation with Desmos solution

Choice B is correct. The area, $A$, of a triangle is given by the formula $A=\frac{1}{2}bh$, where $b$ represents the length of the base of the triangle and $h$ represents its height. It’s given that the area of a triangle is $270$ square centimeters and that the length of the base of this triangle is $12$ centimeters greater than the height of the triangle. Let $x$ represent the height, in centimeters, of the triangle. It follows that the length of the base of the triangle can be expressed as $x+12$. Substituting $270$ for $A$, $x$ for $h$, and $x+12$ for $b$ in the formula $A=\frac{1}{2}bh$ yields $270=\frac{1}{2}(x+12)(x)$, or $270=\frac{1}{2}x(x+12)$. Multiplying both sides of this equation by $2$ yields $540=x(x+12)$. Applying the distributive property on the right-hand side of this equation yields $540=x^2+12x$. Subtracting $540$ from both sides of this equation yields $0=x^2+12x-540$. In factored form, this equation is equivalent to $(x+30)(x-18)=0$. Applying the zero product property, it follows that $x+30=0$ or $x-18=0$. Subtracting $30$ from both sides of the equation $x+30=0$ yields $x=-30$. Adding $18$ to both sides of the equation $x-18=0$ yields $x=18$. Since $x$ represents the height of the triangle, it must be positive. Therefore, the height, in centimeters, of the triangle is $18$.