Respuesta :
Given the system of equations:
[tex]\begin{gathered} f\mleft(x\mright)=6x^2-4x-25 \\ g\mleft(x\mright)=3x-5 \end{gathered}[/tex]To solve the system of equations, we will solve the following equation:
[tex]f(x)=g(x)[/tex]so,
[tex]6x^2-4x-25=3x-5[/tex]Solve the quadratic equation as follows:
[tex]\begin{gathered} 6x^2-4x-25-3x+5=0 \\ 6x^2-7x-20=0 \\ \end{gathered}[/tex]The last equation will be solved using the general formula:
a = 6, b = -7, c = -20
[tex]\begin{gathered} x=\frac{7\pm\sqrt[]{(-7)^2-4\cdot6\cdot(-20)}}{2\cdot6}=\frac{7\pm23}{12} \\ \\ x=\frac{7+23}{12}=2.5 \\ or \\ x=\frac{7-23}{12}=-\frac{4}{3} \end{gathered}[/tex]So, there are two points of intersection for the given system
For each value of (x) substitute into g(x) to find the value of (y)
[tex]\begin{gathered} x=2.5\rightarrow y=3\cdot2.5-5=2.5 \\ x=-\frac{4}{3}\rightarrow y=3\cdot-\frac{4}{3}-5=-9 \end{gathered}[/tex]So, the answer will be, the point of intersections are:
[tex](x,y)=\mleft\lbrace(2.5,2.5\mright);(-\frac{4}{3},-9)\}[/tex]