Solve for `x : 8x^(2) - 2x - 3 = 0`

Answer

To solve the quadratic equation \( 8x^2 - 2x - 3 = 0 \), we can use the middle-term splitting method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 8 \)
- \( b = -2 \)
- \( c = -3 \)

### Step 2: Calculate the product \( ac \)
We need to calculate the product of \( a \) and \( c \):
\[ ac = 8 \times (-3) = -24 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-24\) (the value of \( ac \)) and add to \(-2\) (the value of \( b \)). The numbers that satisfy these conditions are \( 4 \) and \( -6 \) because:
\[ 4 \times (-6) = -24 \quad \text{and} \quad 4 + (-6) = -2 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the equation by splitting the middle term using the numbers we found:
\[ 8x^2 + 4x - 6x - 3 = 0 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (8x^2 + 4x) + (-6x - 3) = 0 \]

### Step 6: Factor out the common terms
Now, we factor out the common terms from each group:
\[ 4x(2x + 1) - 3(2x + 1) = 0 \]

### Step 7: Factor by grouping
Now, we can factor out the common binomial factor \( (2x + 1) \):
\[ (2x + 1)(4x - 3) = 0 \]

### Step 8: Set each factor to zero
Now, we set each factor equal to zero:
1. \( 2x + 1 = 0 \)
2. \( 4x - 3 = 0 \)

### Step 9: Solve for \( x \)
For the first equation:
\[ 2x + 1 = 0 \implies 2x = -1 \implies x = -\frac{1}{2} \]

For the second equation:
\[ 4x - 3 = 0 \implies 4x = 3 \implies x = \frac{3}{4} \]

### Final Solution
The solutions for \( x \) are:
\[ x = -\frac{1}{2} \quad \text{and} \quad x = \frac{3}{4} \]

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