Question

use a graphing utility to find or to approximate the x - intercept(s) of the graph of the function.
$y = 5x^{2}+17x - 12$
graph the function $y = 5x^{2}+17x - 12$. choose the correct graph below.
graphs are shown in $-10,10$ by $-50,50$ viewing rectangles with $xscl = 1$ and $yscl = 5$.
use the graph found in the previous step to find the intercepts, if any. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x - intercept(s) is/are $x=$
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there is no x - intercept

Explanation:

Step1: Recall the formula for x - intercepts

The x - intercepts of the function \(y = ax^{2}+bx + c\) are found by setting \(y = 0\), so we solve the quadratic equation \(5x^{2}+17x - 12=0\).
The quadratic formula for \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a = 5\), \(b = 17\), and \(c=-12\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(17)^{2}-4\times5\times(-12)=289 + 240=529\).

Step2: Apply the quadratic formula

Substitute the values into the quadratic formula:
\(x=\frac{-17\pm\sqrt{529}}{2\times5}=\frac{-17\pm23}{10}\).
For the plus - sign: \(x=\frac{-17 + 23}{10}=\frac{6}{10}=0.6\).
For the minus - sign: \(x=\frac{-17-23}{10}=\frac{-40}{10}=-4\).

Answer:

The x - intercept(s) is/are \(x=-4,0.6\)