What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$?
I notice that this equation is quadratic, so I can try to use the quadratic formula to find the values of $x$ that satisfy it. Alternatively, I can try to factor the equation, but I don't see any obvious factors that would work. So I'll go with the quadratic formula, which says that if I have an equation of the form $ax^2+bx+c=0$, then the solutions are given by $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. In this case, I can compare my equation to the standard form and see that $a=1$, $b=-5$, and $c=5-9=-4$. So I can plug these values into the formula and get $x=\frac{-(-5)\pm\sqrt{(-5)^2-4(1)(-4)}}{2(1)}$. Simplifying this expression, I get $x=\frac{5\pm\sqrt{25+16}}{2}$. Further simplifying, I get $x=\frac{5\pm\sqrt{41}}{2}$. These are the two values of $x$ that satisfy the equation, so to find their sum, I just add them together. I get $x_1+x_2=\frac{5+\sqrt{41}}{2}+\frac{5-\sqrt{41}}{2}$. Notice that the terms with the square roots cancel out, leaving me with $x_1+x_2=\frac{5}{2}+\frac{5}{2}$. So the sum is just $5$. 