https://file.aiquickdraw.com/mathbot/file/17239072019030dc70evg.jpg

solve this math (as shown in pic) with completing square method.

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.

Learn how to solve a math problem using the completing square method with detailed step-by-step instructions. This problem involves quadratic equations and algebraic techniques, suitable for Grades 10-12.

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