The problem requires us to solve the quadratic equation $x^2 - 9x + 20 = 0$ by graphing. This means we need to find the $x$-intercepts of the parabola $y = x^2 - 9x + 20$.

1. Problem Description

The problem requires us to solve the quadratic equation $x^2 - 9x + 20 = 0$ by graphing. This means we need to find the $x$-intercepts of the parabola $y = x^2 - 9x + 20$.

2. Solution Steps

First, consider the function $y = x^2 - 9x + 20$. We are looking for the values of $x$ such that $y=0$.

We can find the vertex of the parabola using the formula $x = -\frac{b}{2a}$, where $a=1$ and $b=-9$.

$x = -\frac{-9}{2(1)} = \frac{9}{2} = 4.5$

Now we can find the corresponding $y$-value by plugging $x=4.5$ into the equation:

$y = (4.5)^2 - 9(4.5) + 20 = 20.25 - 40.5 + 20 = -0.25$

So the vertex of the parabola is $(4.5, -0.25)$.

We can also find the y-intercept by setting $x=0$:

$y = (0)^2 - 9(0) + 20 = 20$

So the y-intercept is $(0, 20)$.

To find the x-intercepts, we set $y = 0$ and solve for $x$:

$x^2 - 9x + 20 = 0$

This quadratic equation can be factored:

$(x - 4)(x - 5) = 0$

Therefore, the solutions are $x = 4$ and $x = 5$.

These are the x-intercepts of the parabola, which are the solutions to the original equation.

3. Final Answer

$x = 4, 5$