what is the positive solution of the equation 1/2x^2 -3x =5? solve by using a table or by graphing. if necessary, round your answer to the nearest hundredth.
To solve the equation 1/2x^2 - 3x = 5, you can either create a table of values or graph the equation to find the positive solution. Let's go through both methods.
Method 1: Creating a Table of Values
To create a table of values, you can choose different values of x and calculate the corresponding values of y. Then you can find the x-value where y = 5.
1. Start by selecting a few values of x. For example, let's choose x = -10, -5, 0, 5, and 10.
2. Substitute each value of x into the equation 1/2x^2 - 3x = 5 to calculate the corresponding y-values.
For x = -10: (1/2(-10)^2) - 3(-10) = 50 + 30 = 80
For x = -5: (1/2(-5)^2) - 3(-5) = 12.5 + 15 = 27.5
For x = 0: (1/2(0)^2) - 3(0) = 0 - 0 = 0
For x = 5: (1/2(5)^2) - 3(5) = 12.5 - 15 = -2.5
For x = 10: (1/2(10)^2) - 3(10) = 50 - 30 = 20
3. Arrange the x and y-values in a table:
| x | y |
| -10 | 80 |
| -5 | 27.5 |
| 0 | 0 |
| 5 | -2.5 |
| 10 | 20 |
4. By looking at the table, you can see that there is no positive value of x for which y equals approximately 5. Therefore, there is no positive solution in this case.
Method 2: Graphing
Graphing the equation will help visualize where the curve intersects the y = 5 line. Here's how to do it:
1. Rearrange the equation to have 0 on one side: 1/2x^2 - 3x - 5 = 0.
2. Plot the graph of the equation y = 1/2x^2 - 3x - 5 (you can use graphing software or a graphing calculator).
3. Draw a horizontal line at y = 5.
4. Locate the points where the curve intersects the y = 5 line.
Upon graphing, you will see that the curve does not intersect the y = 5 line on the positive side of the x-axis. Therefore, there is no positive solution in this case.
In conclusion, by using both the table method and the graphing method, we find that there is no positive solution to the equation 1/2x^2 - 3x = 5.