Q. 30 - Precalculus Enhanced with Graphing Utilities

Question

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$5x^2=13x+6$

Step-by-Step Solution

Answer

The required solution is $x=-\frac{2}{5},3$ and the graph of the solution is shown below:

Step 1. Given information.

We have:

$5x^2=13x+6$

Step 2. Solve the equation algebraically.

The equation is $5x^2=13x+6$.

$5x^2-13x-6=0$

For a quadratic equation of the form $a{x}^2+bx+c=0$ the solutions are:

$x_{1,\hspace{0.17em}2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

For $a=5,\hspace{0.17em}b=-13,\hspace{0.17em}c=-6$

$$\begin{gathered} x_{1,\hspace{0.17em}2}=\frac{-\left(-13\right)\pm \sqrt{{\left(-13\right)}^2-4·\hspace{0.17em}5\left(-6\right)}}{2·\hspace{0.17em}5} \\ {x}_{1,\hspace{0.17em}2}=\frac{-\left(-13\right)\pm \hspace{0.17em}17}{2·\hspace{0.17em}5} \\ x=3,-\frac{2}{5} \end{gathered}$$

Step 3. Verify the solution using a graphing utility.

The solution of the equation is $x=-\frac{2}{5},3$.

Draw the graph of the equation $5x^2=13x+6$ using the graphing utility: