The given equation in the image is:

$\frac{(2x - 1)^2 - (2x + 1)^2}{4x^2 - 1} = -\frac{4}{3}$

Let's solve this step by step using the completing square method.

Step 1: Expand the squared terms

We have two squared terms in the numerator: $(2x - 1)^2 \quad \text{and} \quad (2x + 1)^2$

Expanding each of them:

$(2x - 1)^2 = 4x^2 - 4x + 1$

$(2x + 1)^2 = 4x^2 + 4x + 1$

Step 2: Substitute the expanded forms back into the equation

Substituting the expanded forms:

$\frac{(4x^2 - 4x + 1) - (4x^2 + 4x + 1)}{4x^2 - 1} = -\frac{4}{3}$

Step 3: Simplify the numerator

Simplifying the numerator:

$(4x^2 - 4x + 1) - (4x^2 + 4x + 1) = 4x^2 - 4x + 1 - 4x^2 - 4x - 1$

$= -8x$

Now the equation becomes:

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

Step 4: Cross-multiply to simplify

Cross-multiply to get rid of the fraction:

$-8x \cdot 3 = -4 \cdot (4x^2 - 1)$

Simplifying:

$-24x = -16x^2 + 4$

Step 5: Rearrange the equation

Rearrange all terms to one side to set the equation to zero:

$16x^2 - 24x + 4 = 0$

Step 6: Factor or use the quadratic formula

The equation is now a quadratic equation:

$16x^2 - 24x + 4 = 0$

To solve by completing the square:

1. Divide the equation by 16: $x^2 - \frac{3}{2}x + \frac{1}{4} = 0$

2. Move the constant term to the right side: $x^2 - \frac{3}{2}x = -\frac{1}{4}$

3. Complete the square by adding and subtracting $\left(\frac{3}{4}\right)^2$ on the left side: $x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{9}{16} - \frac{1}{4}$

4. Simplify the right side: $\left(x - \frac{3}{4}\right)^2 = \frac{5}{16}$

5. Take the square root of both sides: $x - \frac{3}{4} = \pm \frac{\sqrt{5}}{4}$

6. Solve for $x$: $x = \frac{3 \pm \sqrt{5}}{4}$

Thus, the solutions are:

$x = \frac{3 + \sqrt{5}}{4} \quad \text{and} \quad x = \frac{3 - \sqrt{5}}{4}$

Would you like further details on this solution or have any other questions?

Follow-Up Questions

1. What is the next step after finding the roots of a quadratic equation?
2. How does completing the square help in solving quadratic equations?
3. Can this problem be solved using a different method, such as the quadratic formula?
4. What does the discriminant tell us about the nature of the roots in a quadratic equation?
5. How would the problem change if the original equation was set equal to a different constant?

Tip

When solving quadratic equations, always check if factoring is possible before moving on to other methods like completing the square or the quadratic formula.