Solve -x^2=4x by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.

To solve the equation -x^2 = 4x by graphing, let's rearrange the equation to bring all terms to one side:

-x^2 - 4x = 0

Now, plot the graph of the equation y = -x^2 - 4x.

To find the roots, we need to identify the x-values where the graph intersects or touches the x-axis. These x-values correspond to the solutions of the equation.

Here's a step-by-step guide to graphing the equation:

Step 1: Create a coordinate plane by drawing the x and y-axis.

Step 2: Determine the range of x-values to plot. To do this, we can either use a graphing calculator or analyze the equation:

For this equation, since the term -x^2 implies a negative quadratic, the graph will be symmetric with its vertex being the highest or lowest point. In this case, the vertex corresponds to the highest point since the coefficient of -x^2 is negative (-1). By symmetry, the graph will continue indefinitely towards both sides.

To find the x-value of the vertex, we can use the formula x = -b/2a. In this case, a = -1 and b = -4.

x = -(-4)/(2*(-1)) = 2

So, the vertex is located at (2, y).

Step 3: Select some x-values to the left and right of the vertex (2) and calculate their corresponding y-values.

Let's choose x = 0, 1, and 3:

For x = 0:
y = -(0)^2 - 4(0) = 0

For x = 1:
y = -(1)^2 - 4(1) = -1 - 4 = -5

For x = 3:
y = -(3)^2 - 4(3) = -9 - 12 = -21

Step 4: Plot the points (x, y) calculated in Step 3.

The plotted points are:
(0, 0), (1, -5), (2, y), (3, -21).

Step 5: Draw a smooth curve passing through the plotted points. The graph should be a downward-opening parabola, symmetrical about the x-axis.

Step 6: Observe where the graph intersects or touches the x-axis. These points determine the roots of the equation.

Based on the graph, we can see that the graph intersects the x-axis between x = -4 and x = -3, and between x = 3 and x = 4.

Therefore, the consecutive integers between which the roots are located are -4 and -3, and 3 and 4.

Hence, the solutions to -x^2 = 4x are -4, -3, 3, and 4.

To solve the equation -x^2 = 4x by graphing, we need to plot the graph of the equation and find the x-values where the graph intersects the x-axis. These x-values will be the solutions to the equation.

Step 1: Rewrite the equation in the form f(x) = 0.
-x^2 = 4x can be rewritten as -x^2 - 4x = 0.

Step 2: Plot the graph
To plot the graph, we create a table of x and f(x) values. Choose a range of x-values and substitute them into the equation to find the corresponding f(x) values. Let's choose x values from -5 to 5, for example.

x | f(x)
---------
-5 | -25
-4 | -16
-3 | -9
-2 | -4
-1 | -1
0 | 0
1 | 3
2 | 4
3 | 3
4 | 0
5 | -5

Now, plot the points (x, f(x)) on a graph.

Step 3: Analyze the graph
Look for the points where the graph crosses or touches the x-axis. These are the solutions to the equation -x^2 = 4x, since the value of f(x) is 0 at those points.

On the graph, we see that the graph intersects the x-axis at x = -4 and x = 0. Therefore, -4 and 0 are the solutions to the equation -x^2 = 4x.

Answer: The solutions to the equation -x^2 = 4x are x = -4 and x = 0.

make a table of values and plot those points.

Observe where the graph crosses the x-axis.