Certainly! Let's solve the given equation step by step.
Given equation:
[tex]\[
\frac{3}{2x^2} = \frac{1}{x} - \frac{1}{8}
\][/tex]
Step 1: Eliminate fractions by finding a common denominator.
The least common multiple (LCM) of [tex]\( 2x^2 \)[/tex], [tex]\( x \)[/tex], and [tex]\( 8 \)[/tex] is [tex]\( 8x^2 \)[/tex].
Multiply both sides of the equation by [tex]\( 8x^2 \)[/tex] to clear the fractions:
[tex]\[
8x^2 \cdot \frac{3}{2x^2} = 8x^2 \cdot \left( \frac{1}{x} - \frac{1}{8} \right)
\][/tex]
Step 2: Simplify the equation after multiplying by [tex]\( 8x^2 \)[/tex]:
[tex]\[
8x^2 \cdot \frac{3}{2x^2} = 8x^2 \cdot \frac{1}{x} - 8x^2 \cdot \frac{1}{8}
\][/tex]
This reduces to:
[tex]\[
4 \cdot 3 = 8x - x^2
\][/tex]
So, we get:
[tex]\[
12 = 8x - x^2
\][/tex]
Step 3: Rearrange the equation to form a standard quadratic equation:
Move all terms to one side to set the equation to 0,
[tex]\[
x^2 - 8x + 12 = 0
\][/tex]
Step 4: Solve the quadratic equation using the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the equation [tex]\( x^2 - 8x + 12 = 0 \)[/tex], the coefficients are:
[tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 12 \)[/tex].
Substitute these values into the quadratic formula:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{8 \pm \sqrt{64 - 48}}{2}
\][/tex]
[tex]\[
x = \frac{8 \pm \sqrt{16}}{2}
\][/tex]
[tex]\[
x = \frac{8 \pm 4}{2}
\][/tex]
So the solutions are:
[tex]\[
x = \frac{8 + 4}{2} = \frac{12}{2} = 6
\][/tex]
and
[tex]\[
x = \frac{8 - 4}{2} = \frac{4}{2} = 2
\][/tex]
Therefore, the solutions to the equation [tex]\( \frac{3}{2x^2} = \frac{1}{x} - \frac{1}{8} \)[/tex] are:
[tex]\[
\boxed{2 \text{ and } 6}
\][/tex]
