The problem asks us to solve the quadratic equation $8x^2 - 35 = -18x$ for $x$ using the quadratic formula, and express the answers as simplified fractions.

1. Problem Description

The problem asks us to solve the quadratic equation $8x^2 - 35 = -18x$ for $x$ using the quadratic formula, and express the answers as simplified fractions.

2. Solution Steps

First, we need to rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$.

Adding $18x$ to both sides, we get $8x^2 + 18x - 35 = 0$.

Now we can identify the coefficients: $a = 8$, $b = 18$, and $c = -35$.

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in the values for $a$, $b$, and $c$, we get:

$x = \frac{-18 \pm \sqrt{18^2 - 4(8)(-35)}}{2(8)}$

$x = \frac{-18 \pm \sqrt{324 + 1120}}{16}$

$x = \frac{-18 \pm \sqrt{1444}}{16}$

Since $\sqrt{1444} = 38$, we have:

$x = \frac{-18 \pm 38}{16}$

Now we find the two possible values for $x$:

$x_1 = \frac{-18 + 38}{16} = \frac{20}{16} = \frac{5}{4}$

$x_2 = \frac{-18 - 38}{16} = \frac{-56}{16} = \frac{-14}{4} = -\frac{7}{2}$

3. Final Answer

The two possible values of $x$ are $\frac{5}{4}$ and $-\frac{7}{2}$.