Solve the following equation : `(x^(2)-3.5 x + 1.5)/(x^(2)-x-6)=0`

Answer

To solve the equation \(\frac{x^2 - 3.5x + 1.5}{x^2 - x - 6} = 0\), we can follow these steps:

### Step 1: Understand the Equation
The equation is a fraction set to zero. A fraction is equal to zero when its numerator is zero (as long as the denominator is not zero).

### Step 2: Set the Numerator to Zero
We need to set the numerator equal to zero:
\[ x^2 - 3.5x + 1.5 = 0 \]

### Step 3: Eliminate Decimals
To eliminate the decimal, we can multiply the entire equation by 2:
\[ 2(x^2 - 3.5x + 1.5) = 0 \implies 2x^2 - 7x + 3 = 0 \]

### Step 4: Factor the Quadratic Equation
Next, we will factor the quadratic equation \(2x^2 - 7x + 3 = 0\). We look for two numbers that multiply to \(2 \cdot 3 = 6\) and add to \(-7\). The numbers \(-6\) and \(-1\) work:
\[ 2x^2 - 6x - x + 3 = 0 \]
Grouping gives us:
\[ 2x(x - 3) - 1(x - 3) = 0 \]
Factoring out \((x - 3)\):
\[ (2x - 1)(x - 3) = 0 \]

### Step 5: Solve for x
Now, we can set each factor to zero:
1. \(2x - 1 = 0\) gives \(x = \frac{1}{2}\)
2. \(x - 3 = 0\) gives \(x = 3\)

### Step 6: Check the Denominator
We must ensure that the denominator does not equal zero. The denominator is:
\[ x^2 - x - 6 = 0 \]
Factoring this gives:
\[ (x - 3)(x + 2) = 0 \]
Thus, \(x = 3\) or \(x = -2\).

Since \(x = 3\) makes the denominator zero, we discard this solution.

### Step 7: Final Solution
The only valid solution is:
\[ x = \frac{1}{2} \]