Select the correct answer.

Solve the following equation for [tex]x[/tex]:

[tex]80x^2 + 92x - 84 = 0[/tex]

A. [tex]x = \frac{4}{3}, \frac{7}{5}[/tex]
B. [tex]x = -\frac{7}{4}, \frac{3}{5}[/tex]
C. [tex]x = -\frac{7}{5}, -\frac{4}{3}[/tex]
D. [tex]x = -\frac{3}{5}, \frac{7}{4}[/tex]



Answer :

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].