To solve the quadratic equation [tex]\( 80x^2 + 92x - 84 = 0 \)[/tex], we can use the quadratic formula. The quadratic formula states:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients from the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
For the given equation [tex]\( 80x^2 + 92x - 84 = 0 \)[/tex]:
- [tex]\( a = 80 \)[/tex]
- [tex]\( b = 92 \)[/tex]
- [tex]\( c = -84 \)[/tex]
First, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 92^2 - 4 \cdot 80 \cdot (-84)
\][/tex]
[tex]\[
\Delta = 8464 + 26880
\][/tex]
[tex]\[
\Delta = 35344
\][/tex]
Next, we calculate the two possible values for [tex]\( x \)[/tex] using the quadratic formula:
[tex]\[
x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\( b \)[/tex], [tex]\( \Delta \)[/tex], and [tex]\( a \)[/tex] into the formula:
[tex]\[
x_1 = \frac{-92 + \sqrt{35344}}{2 \cdot 80}
\][/tex]
[tex]\[
x_2 = \frac{-92 - \sqrt{35344}}{2 \cdot 80}
\][/tex]
Simplifying these expressions, we get:
[tex]\[
x_1 = \frac{-92 + 188}{160} = \frac{96}{160} = \frac{3}{5}
\][/tex]
[tex]\[
x_2 = \frac{-92 - 188}{160} = \frac{-280}{160} = -\frac{7}{4}
\][/tex]
So the solutions to the equation [tex]\( 80x^2 + 92x - 84 = 0 \)[/tex] are [tex]\( x = \frac{3}{5} \)[/tex] and [tex]\( x = -\frac{7}{4} \)[/tex].
Thus, the correct answer is [tex]\( \boxed{x = -\frac{7}{4}, \frac{3}{5}} \)[/tex].
