Question

X^3-4x^2=-x-6
`{x}^(3) - 4 {x}^(2) = - x - 6`what are the solutions of the equation?

Answer 1

Answer: x= -1, x = 2 and x = 3

In order to graph out the equation given, we need to create a table of values that will guide us.

This table of values shall contain values of x and corresponding values of y.

For the values of x, we shall use integers ranging from -2 to +4.

While for the values of y, we shall plug in this chosen x-coordinates into the equation given.

`\begin{gathered} x^3-4x^2=-x-6 \\ We\text{ can re-write this equation as:} \\ y=x^3-4x^2+x+6 \\ \\ \end{gathered}`

Now, to get the y-values and hence create the table of values which will be used to plot the graph:

`\begin{gathered} \text{when x= -2;} \\ y=(-2)^3-4(-2)^2+(-2)+6=-20 \\ \text{when x=-1;} \\ y=(-1)^3-4(-1)^2+(-1)+6=0 \\ \text{when x= 0}; \\ y=(0)^3-4(0)^2+(0)+6=6 \\ \text{when x = 1}; \\ y=(1)^3-4(1)^2+(1)+6=4 \\ \text{when x=2}; \\ y=(2)^3-4(2)^2+(2)+6=0 \\ \text{when x = 3;} \\ y=(3)^3-4(3)^2+(3)+6=0 \\ \text{when x = 4;} \\ y=(4)^3-4(4)^2+4+6=10 \end{gathered}`

Thus, the coordinates of the equation are:

(x, y):

(-2, -20), (-1, 0), (0, 1), (1, 4), (2, 0), (3, 0), (4, 10)

We can therefore create a table of values for the calculated coordinates of the equation

Now with this table of values, you can plot the graph. The picture of the plotted graph is shown below:

The solutions to the equation are where the graph crosses the x-axis.

By reading the graph above, the solutions are:

x= -1, x = 2 and x = 3

Therefore, the final answer is: x= -1, x = 2 and x = 3