The given system of equations is:
[tex]\begin{gathered} y=6x+1\text{ Eq.(1)} \\ y=2x^2-10x-2\text{ Eq.(2)} \end{gathered}[/tex]
As y=y then we can equal both equations and solve for x as follows:
[tex]\begin{gathered} y=y \\ 6x+1=2x^2-10x-2 \\ \text{Subtract 6x from both sides} \\ 6x+1-6x=2x^2-10x-2-6x \\ 1=2x^2-16x-2 \\ \text{Subtract 1 from both sides} \\ 1-1=2x^2-16x-2-1 \\ 0=2x^2-16x-3 \end{gathered}[/tex]
Now, we have an equation in the form:
[tex]ax^2+bx+c=0^{}[/tex]
Where a=2, b=-16 and c=-3, we can apply the quadratic formula:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]
Replace the a, b and c values and solve to find the x-values:
[tex]\begin{gathered} x=\frac{-(-16)\pm\sqrt[]{(-16)^2-4(2)(-3)}}{2(2)} \\ x=\frac{16\pm\sqrt[]{256+24}}{4} \\ x=\frac{16\pm\sqrt[]{280}}{4} \\ x=\frac{16\pm\sqrt[]{4\cdot70}}{4} \\ x=\frac{16\pm2\sqrt[]{70}}{4} \\ \text{Then:} \\ x1=\frac{16+2\sqrt[]{70}}{4}=\frac{16}{4}+\frac{2\sqrt[]{70}}{4}=4+4.18=8.18 \\ x2=\frac{16-2\sqrt[]{70}}{4}=\frac{16}{4}-\frac{2\sqrt[]{70}}{4}=4-4.18=-0.18 \end{gathered}[/tex]
Then the x-values are: 8.18 and -0.18, and its multiplication is:
[tex]8.18\cdot-0.18=-1.5[/tex]
Answer: -1.5
