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

Answer

To solve the quadratic equation \(8x^2 - 22x - 21 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 8\)
- \(b = -22\)
- \(c = -21\)

### Step 2: Factor the quadratic equation
We need to factor the quadratic expression. We look for two numbers that multiply to \(a \cdot c = 8 \cdot (-21) = -168\) and add up to \(b = -22\).

The two numbers that satisfy these conditions are \(-28\) and \(6\) because:
- \(-28 \cdot 6 = -168\)
- \(-28 + 6 = -22\)

### Step 3: Rewrite the equation
We can rewrite the equation by splitting the middle term:
\[ 8x^2 - 28x + 6x - 21 = 0 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (8x^2 - 28x) + (6x - 21) = 0 \]

### Step 5: Factor by grouping
Now we factor out the common terms from each group:
\[ 4x(2x - 7) + 3(2x - 7) = 0 \]

### Step 6: Factor out the common binomial
We can factor out the common binomial \((2x - 7)\):
\[ (4x + 3)(2x - 7) = 0 \]

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

### Step 8: Solve for \(x\)
Now we solve each equation for \(x\):
1. From \(4x + 3 = 0\):
\[ 4x = -3 \implies x = -\frac{3}{4} \]

2. From \(2x - 7 = 0\):
\[ 2x = 7 \implies x = \frac{7}{2} \]

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

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