Solve each of the following equations by using the method of completing the square:
`8x^(2)-14x-15=0`

Answer

To solve the equation \( 8x^2 - 14x - 15 = 0 \) using the method of completing the square, we will follow these steps:

### Step 1: Divide the entire equation by the coefficient of \( x^2 \)
We start by dividing the whole equation by 8 to make the coefficient of \( x^2 \) equal to 1.

\[ \frac{8x^2}{8} - \frac{14x}{8} - \frac{15}{8} = 0 \]

This simplifies to:

\[ x^2 - \frac{14}{8}x - \frac{15}{8} = 0 \]

### Step 2: Simplify the coefficients
Next, we simplify \( \frac{14}{8} \) to \( \frac{7}{4} \):

\[ x^2 - \frac{7}{4}x - \frac{15}{8} = 0 \]

### Step 3: Move the constant term to the right side
Now, we move the constant term to the right side of the equation:

\[ x^2 - \frac{7}{4}x = \frac{15}{8} \]

### Step 4: Complete the square
To complete the square, we need to add and subtract the square of half the coefficient of \( x \). The coefficient of \( x \) is \( -\frac{7}{4} \), so half of this is \( -\frac{7}{8} \). Now we square it:

\[ \left(-\frac{7}{8}\right)^2 = \frac{49}{64} \]

We add this value to both sides:

\[ x^2 - \frac{7}{4}x + \frac{49}{64} = \frac{15}{8} + \frac{49}{64} \]

### Step 5: Simplify the right side
To add the fractions on the right side, we need a common denominator. The common denominator of 8 and 64 is 64. We convert \( \frac{15}{8} \):

\[ \frac{15}{8} = \frac{15 \times 8}{8 \times 8} = \frac{120}{64} \]

Now we can add:

\[ \frac{120}{64} + \frac{49}{64} = \frac{169}{64} \]

### Step 6: Rewrite the left side as a square
Now, we can rewrite the left side as a perfect square:

\[ \left(x - \frac{7}{8}\right)^2 = \frac{169}{64} \]

### Step 7: Take the square root of both sides
Taking the square root of both sides gives:

\[ x - \frac{7}{8} = \pm \frac{13}{8} \]

### Step 8: Solve for \( x \)
Now we solve for \( x \) by considering both the positive and negative cases.

1. **Positive case**:
\[ x - \frac{7}{8} = \frac{13}{8} \]
\[ x = \frac{13}{8} + \frac{7}{8} = \frac{20}{8} = \frac{5}{2} \]

2. **Negative case**:
\[ x - \frac{7}{8} = -\frac{13}{8} \]
\[ x = -\frac{13}{8} + \frac{7}{8} = -\frac{6}{8} = -\frac{3}{4} \]

### Final Solution
Thus, the solutions to the equation \( 8x^2 - 14x - 15 = 0 \) are:

\[ x = \frac{5}{2} \quad \text{and} \quad x = -\frac{3}{4} \]

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