You can solve a polynomial equation by graphing each side of the equation separately on the same coordinate system. Then, to find the solution, just get the x-values at the point of intersection.

**Example**

Solve x^{3} + 3x^{2} = x + 3 by graphing

**Step 1**

Use a graphing calculator to graph y_{1} = x^{3} + 3x^{2} and y_{2} = x + 3 on the same screen. A portion of the graph is shown below. y_{1} is the graph shown in red and y_{2} is the graph shown in green.

**Step 2**

You can use either the graph above to locate the points of intersection or you can use the intersect feature in your calculator to find the x-values at the points of intersection.

The points of intersection are shown on the graph above with blue dots. The x-values give the solution. Therefore, the solutions are -3, -1, and 1.

**Check**

Show that -3, -1, and 1 are indeed solutions by plugging each value in the original solution.

x = -3

x^{3} + 3x^{2} = x + 3

(-3)^{3} + 3(-3)^{2} = -3 + 3

-27 + 3(9) = 0

-27 + 27 = 0

0 = 0

x = -1

(-1)^{3} + 3(-1)^{2} = -1 + 3

(-1)^{3} + 3(-1)^{2} = -1 + 3

-1 + 3(1) = 2

-1 + 3 = 2

2 = 2

x = 1

x^{3} + 3x^{2} = x + 3

(1)^{3} + 3(1)^{2} = 1 + 3

1 + 3(1) = 4

1 + 3 = 4

4 = 4