To solve the equation provided in the image:

$1 - \frac{3}{4x} + \frac{1}{8x^2} = 0$

Let's go through the steps.

Step 1: Eliminate Fractions

Multiply both sides by $8x^2$ to clear the denominators:

$8x^2 \cdot 1 - 8x^2 \cdot \frac{3}{4x} + 8x^2 \cdot \frac{1}{8x^2} = 0$

This simplifies to:

$8x^2 - 6x + 1 = 0$

Step 2: Solve the Quadratic Equation

The equation $8x^2 - 6x + 1 = 0$ is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 8$, $b = -6$, and $c = 1$.

To solve for $x$, use the quadratic formula:

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

Substitute the values of $a$, $b$, and $c$:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 8 \cdot 1}}{2 \cdot 8}$

Simplify:

$x = \frac{6 \pm \sqrt{36 - 32}}{16}$

$x = \frac{6 \pm \sqrt{4}}{16}$

$x = \frac{6 \pm 2}{16}$

This gives two solutions:

1. $x = \frac{6 + 2}{16} = \frac{8}{16} = \frac{1}{2}$
2. $x = \frac{6 - 2}{16} = \frac{4}{16} = \frac{1}{4}$

Conclusion

The correct answer is $x_1 = \frac{1}{2}$ and $x_2 = \frac{1}{4}$, which corresponds to option (d).

Would you like further details on any step, or do you have additional questions?

Related Questions

1. How do we verify the solutions for a quadratic equation?
2. What are alternative methods for solving quadratic equations?
3. How can we derive the quadratic formula?
4. What are the applications of quadratic equations in real-world problems?
5. Why is clearing fractions often useful in solving equations?

Tip

When dealing with equations with fractions, consider multiplying through by the least common multiple (LCM) of the denominators to simplify the process.