Solve for x by fractorisation
`8x ^(2) - 22x - 21=0`

Answer

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

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

### Step 2: Calculate the product of \(a\) and \(c\)
We need to find the product \(ac\):
\[ ac = 8 \times (-21) = -168 \]

### Step 3: 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:
\[ -28 \times 6 = -168 \quad \text{and} \quad -28 + 6 = -22 \]

### Step 4: Rewrite the middle term using the two numbers
Now, we can rewrite the equation by splitting the middle term:
\[ 8x^2 - 28x + 6x - 21 = 0 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (8x^2 - 28x) + (6x - 21) = 0 \]

### Step 6: Factor out the common factors from each group
From the first group \(8x^2 - 28x\), we can factor out \(4x\):
\[ 4x(2x - 7) \]
From the second group \(6x - 21\), we can factor out \(3\):
\[ 3(2x - 7) \]

### Step 7: Combine the factored terms
Now we can write the equation as:
\[ 4x(2x - 7) + 3(2x - 7) = 0 \]
This can be factored further:
\[ (2x - 7)(4x + 3) = 0 \]

### Step 8: Set each factor to zero and solve for \(x\)
Now, we set each factor to zero:
1. \(2x - 7 = 0\)
\[ 2x = 7 \implies x = \frac{7}{2} \]
2. \(4x + 3 = 0\)
\[ 4x = -3 \implies x = -\frac{3}{4} \]

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

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