Solve each of the following equatins :
`x=8x^(2)-22x-21=0`

Answer

To solve the equation \( 8x^2 - 22x - 21 = 0 \), we will use the method of factoring. Here are the steps:

### Step 1: Write the equation in standard form
The equation is already in standard form:
\[ 8x^2 - 22x - 21 = 0 \]

### Step 2: Identify \( a \), \( b \), and \( c \)
From the equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 8 \)
- \( b = -22 \)
- \( c = -21 \)

### Step 3: Calculate the product \( ac \)
Calculate \( ac \):
\[ ac = 8 \times (-21) = -168 \]

### Step 4: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-168\) and add to \(-22\). The numbers are \(-28\) and \(6\) because:
\[ -28 \times 6 = -168 \]
\[ -28 + 6 = -22 \]

### Step 5: Rewrite the middle term using the two numbers
We can rewrite the equation by splitting the middle term:
\[ 8x^2 - 28x + 6x - 21 = 0 \]

### Step 6: Factor by grouping
Group the terms:
\[ (8x^2 - 28x) + (6x - 21) = 0 \]
Factor out the common factors:
\[ 4x(2x - 7) + 3(2x - 7) = 0 \]

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

### Step 8: Set each factor to zero
Set each factor to zero:
1. \( 2x - 7 = 0 \)
2. \( 4x + 3 = 0 \)

### Step 9: Solve for \( x \)
For the first equation:
\[ 2x - 7 = 0 \]
\[ 2x = 7 \]
\[ x = \frac{7}{2} \]

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

### Final Solution
The solutions to the equation \( 8x^2 - 22x - 21 = 0 \) are:
\[ x = \frac{7}{2} \quad \text{and} \quad x = -\frac{3}{4} \]

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