Answer
$(2.5,7.25)$, $(3,8)$
Work Step by Step
Given $$
\begin{cases}
y=x^2-4 x+11 \\
y=-x^2+7 x-4.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$. $$
\begin{aligned}
x^2-4 x+11 & =-x^2+7 x-4 \\
x^2+x^2-4 x-7 x & =-4-11 \\
2 x-11 x & =-15 \\
2 x-11 x+15 & =0.
\end{aligned}
$$ Solve the equation: $$
\begin{aligned}
& a=2 \\
& b=-11 \\
& c=15
\end{aligned}
$$ $$
\begin{aligned}
x& =\frac{-(-11) \pm \sqrt{(-11)^2-4 \cdot 2 \cdot 15}}{2 \cdot 2} \\
x& =\frac{11 \pm 1}{4}\\
x_1&=\frac{11 +1}{4}= 3 \\
x_2&=\frac{11 - 1}{4}= \frac{5}{2}
\end{aligned}
$$ Find the corresponding $y$ values using either of the given equations. $$
\begin{aligned}
y_1&=3^2-4\cdot 3+11\\
& = 8\\
y_2&=2.5^2-4\cdot 2.5+11\\
& = 7.25
\end{aligned}
$$ Plot the two functions in the same window to check the solution(s).
The solution is $(2.5,7.25)$ and $(3,8)$.
