Question

find the solution set.
$8x^{2}+5x - 3 = 0$
separate the two values with a comma.

Explanation:

Step1: Identify quadratic form

Let $t = x^2$, so equation becomes $8t^2 + 5t - 3 = 0$

Step2: Apply quadratic formula

Quadratic formula: $t = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a=8, b=5, c=-3$

$$\begin{align*} t &= \frac{-5 \pm \sqrt{5^2 - 4(8)(-3)}}{2(8)}\\ &= \frac{-5 \pm \sqrt{25 + 96}}{16}\\ &= \frac{-5 \pm \sqrt{121}}{16}\\ &= \frac{-5 \pm 11}{16} \end{align*}$$

Step3: Solve for t values

First root: $t = \frac{-5 + 11}{16} = \frac{6}{16} = \frac{3}{8}$
Second root: $t = \frac{-5 - 11}{16} = \frac{-16}{16} = -1$

Step4: Solve for x from t

For $t = \frac{3}{8}$: $x^2 = \frac{3}{8} \implies x = \pm\sqrt{\frac{3}{8}} = \pm\frac{\sqrt{6}}{4}$
For $t = -1$: $x^2 = -1$ has no real solutions

Answer:

$\frac{\sqrt{6}}{4}, -\frac{\sqrt{6}}{4}$