Find the sum of all the values of $x$ that satisfy the system of equations $y=|x^2-6x+5|$ and $y=\frac{29}{4}-2x$.

Answer 1

First we can examine the graph of $y=|x^2-6x+5|$ to get an idea of how it behaves.

[asy]
Label f;

f.p=fontsize(4);

xaxis(-5,7,Ticks(f, 1.0));

yaxis(-3,5,Ticks(f, 1.0));
real f(real x)

{

return abs(x^2-6*x+5);

}

draw(graph(f,-3,5),blue);

[/asy]

Note that $y=|x^2-6x+5|$ is just the graph of $y=x^2-6x+5$ reflected across the $x$-axis for $x\in(1,5)$, so to find the solutions to the system, we must find the $x$ such that $$x^2-6x+5=\frac{29}{4}-2x\qquad\Rightarrow\qquad x^2-4x-\frac{11}{4}=0.$$We can factor the left hand side and then apply the quadratic formula to get that $$x=\frac{4\pm\sqrt{16+11}}{2}=\frac{4\pm\sqrt{27}}{2}.$$Hence, the two solutions are $x=2+\sqrt{27}/2$ and $x=2-\sqrt{27}/2$. The sum of the two solutions is $\boxed{4}$.