Solve the following quations by using qardratic formula:
`(x-2)/(x+2)+(x+2)/(x-1)=4`

Answer

To solve the equation \(\frac{x-2}{x+2} + \frac{x+2}{x-1} = 4\) using the quadratic formula, we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the fractions on the left side is \((x+2)(x-1)\). Therefore, we can rewrite the equation as:

\[ \frac{(x-2)(x-1) + (x+2)(x+2)}{(x+2)(x-1)} = 4 \]

### Step 2: Cross-multiply
Cross-multiplying gives us:

\[ (x-2)(x-1) + (x+2)(x+2) = 4(x+2)(x-1) \]

### Step 3: Expand both sides
Now, we will expand both sides of the equation:

Left side:
\[ (x-2)(x-1) = x^2 - x - 2x + 2 = x^2 - 3x + 2 \]
\[ (x+2)(x+2) = x^2 + 4x + 4 \]
Combining these:
\[ x^2 - 3x + 2 + x^2 + 4x + 4 = 2x^2 + x + 6 \]

Right side:
\[ 4(x+2)(x-1) = 4(x^2 + 2x - x - 2) = 4(x^2 + x - 2) = 4x^2 + 4x - 8 \]

### Step 4: Set the equation to zero
Now we set the equation to zero by moving all terms to one side:

\[ 2x^2 + x + 6 - (4x^2 + 4x - 8) = 0 \]
This simplifies to:
\[ 2x^2 + x + 6 - 4x^2 - 4x + 8 = 0 \]
\[ -2x^2 - 3x + 14 = 0 \]
Multiplying through by -1 gives:
\[ 2x^2 + 3x - 14 = 0 \]

### Step 5: Identify coefficients
In the quadratic equation \(ax^2 + bx + c = 0\), we have:
- \(a = 2\)
- \(b = 3\)
- \(c = -14\)

### Step 6: Use the quadratic formula
The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-14)}}{2 \cdot 2} \]

### Step 7: Calculate the discriminant
Calculating the discriminant:

\[ b^2 - 4ac = 9 + 112 = 121 \]

### Step 8: Substitute back into the formula
Now substituting back:

\[ x = \frac{-3 \pm \sqrt{121}}{4} \]
\[ x = \frac{-3 \pm 11}{4} \]

### Step 9: Solve for the two possible values of \(x\)
Calculating the two possible values:

1. \(x = \frac{-3 + 11}{4} = \frac{8}{4} = 2\)
2. \(x = \frac{-3 - 11}{4} = \frac{-14}{4} = -\frac{7}{2}\)

### Final Answer
The solutions to the equation are:

\[ x = 2 \quad \text{and} \quad x = -\frac{7}{2} \]

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