Question

use the quadratic formula to solve. express your answer in simplest form.
$4x^2 + 9x - 7 = 2$

Explanation:

Step1: Rewrite the equation in standard form

First, we need to rewrite the equation \(4x^{2}+9x - 7=2\) in the standard quadratic form \(ax^{2}+bx + c = 0\). Subtract 2 from both sides:
\(4x^{2}+9x-7 - 2=0\)
\(4x^{2}+9x - 9 = 0\)
Here, \(a = 4\), \(b=9\), and \(c=- 9\).

Step2: Recall the quadratic formula

The quadratic formula is given by \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

Step3: Substitute the values of \(a\), \(b\), and \(c\) into the formula

Substitute \(a = 4\), \(b = 9\), and \(c=-9\) into the quadratic formula:
First, calculate the discriminant \(D=b^{2}-4ac=(9)^{2}-4\times4\times(-9)\)
\(D = 81+144=225\)
Then, \(x=\frac{-9\pm\sqrt{225}}{2\times4}=\frac{-9\pm15}{8}\)

Step4: Find the two solutions

We have two cases:
Case 1: \(x=\frac{-9 + 15}{8}=\frac{6}{8}=\frac{3}{4}\)
Case 2: \(x=\frac{-9-15}{8}=\frac{-24}{8}=- 3\)

Answer:

\(x=\frac{3}{4}\) or \(x=-3\)