Question

Use a graphing calculator to approximate the solution of the equation. $$ \frac{5}{4} x^{2}+15 x+40=0 $$

Short answer

The solutions to the equation \( \frac{5}{4} x^{2}+15 x+40=0 \) can be approximated using a graphing calculator. The solution involves graphing the equation and finding the x-intercepts. The exact solution may vary depending on the precision of the calculator, but will be approximately the x-values of the points where the graph intersects the x-axis.

Step-by-step solution

Enter the equation in the calculator

First, the given quadratic equation \( \frac{5}{4}x^{2}+15x+40=0 \) is entered into the graphing calculator. Ensure you are in the graphing mode of the calculator, then enter the quadratic equation in the form of \( y = \frac{5}{4}x^{2}+15x+40 \). The right side of the equation is considered as \(y\). After the equation is entered, graph it.

Intercepts calculation

Observe the graph for the points where it intersects the x-axis. These points are the solutions for the given equation. Using the graph, determine the x-intercepts. The x-intercepts are the points where the graph intersects the x-axis, i.e., the points where \(y = 0\). Now, activate the 'intersect' function from the graph's menu on the calculator.

Approximating the solution

While keeping the cursor on the x-intercepts, hit the 'Enter' button. The calculator will now display the x-coordinates of these points. These x-values are the approximate solutions to the given quadratic equation.