Exercise 32 Page 495

We will solve the equation by graphing each side of the equation. The $x\text{-}$coordinates of the points of intersection of the graphs are the solutions of the equation. Let's graph the functions related to the left-hand side and the right-hand side of the equation one at a time.

Left-Hand Side

We will start by graphing the function $y=x^2.$ It is the parent quadratic function. The vertex of the graph of $y=x^2$ is $(0,0)$ and its axis of symmetry is $x=0,$ or the $y\text{-}$axis. Next, let's make a table of values to help us with graphing.

$x$	$y=x^2$
$1$	$1$
$3$	$9$
$5$	$25$

Let's plot these points and their reflection across the axis of symmetry.

Finally, we will connect the points with a smooth curve.

Right-Hand Side

Now we will graph the linear function $y=6x-5.$ Let's rewrite the equation of the second function in slope-intercept form. We will also highlight the slope $m$ and the $y\text{-}$intercept $b.$ Plot the $y\text{-}$intercept and use the slope to plot another point. Then we will connect the points with a straight edge.

Connecting the Graphs

The last step is connecting the graphs and determining their points of intersection. Recall that the $x\text{-}$coordinates of the points of intersection are the solutions of the given equation.

The points of intersection of the graphs are $(1,1)$ and $(5,25).$ Therefore, $x=1$ and $x=5$ are the solutions of the equation $x^2=6x-5.$