Find the range of the expression `(x^(2) + 34x - 71)/(x^(2) + 2x -7)`, if x is a real.

Answer

To find the range of the expression \(\frac{x^2 + 34x - 71}{x^2 + 2x - 7}\), we can follow these steps:

### Step 1: Set the expression equal to \(y\)
Let:
\[ y = \frac{x^2 + 34x - 71}{x^2 + 2x - 7} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ y(x^2 + 2x - 7) = x^2 + 34x - 71 \]
Rearranging this, we have:
\[ yx^2 + 2yx - 7y - x^2 - 34x + 71 = 0 \]

### Step 3: Combine like terms
Combine the terms to form a quadratic equation in \(x\):
\[ (y - 1)x^2 + (2y - 34)x + (71 - 7y) = 0 \]

### Step 4: Apply the condition for real roots
For \(x\) to be a real number, the discriminant \(D\) of this quadratic must be greater than or equal to zero:
\[ D = (2y - 34)^2 - 4(y - 1)(71 - 7y) \geq 0 \]

### Step 5: Expand the discriminant
Calculating \(D\):
\[ D = (2y - 34)^2 - 4[(y - 1)(71 - 7y)] \]
Expanding this:
\[ D = (2y - 34)^2 - 4[(y \cdot 71 - 7y^2 - 71 + 7y)] \]
\[ D = (2y - 34)^2 - 4(71y - 7y^2 - 71 + 7y) \]
\[ D = (2y - 34)^2 - 4(-7y^2 + 78y - 71) \]

### Step 6: Simplify the expression
Now, simplifying the discriminant:
\[ D = 4y^2 - 136y + 1156 + 28y^2 - 312y + 284 \]
Combining like terms:
\[ D = 32y^2 - 448y + 1440 \]

### Step 7: Set the discriminant greater than or equal to zero
We need:
\[ 32y^2 - 448y + 1440 \geq 0 \]

### Step 8: Solve the quadratic inequality
Dividing the entire inequality by 16:
\[ 2y^2 - 28y + 90 \geq 0 \]
Now, we can find the roots of the quadratic equation:
\[ 2y^2 - 28y + 90 = 0 \]
Using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{28 \pm \sqrt{(-28)^2 - 4 \cdot 2 \cdot 90}}{2 \cdot 2} \]
Calculating the discriminant:
\[ 784 - 720 = 64 \]
So,
\[ y = \frac{28 \pm 8}{4} \]
Calculating the roots:
\[ y_1 = \frac{36}{4} = 9, \quad y_2 = \frac{20}{4} = 5 \]

### Step 9: Determine the intervals
The quadratic opens upwards (since the coefficient of \(y^2\) is positive), so the expression \(2y^2 - 28y + 90 \geq 0\) is satisfied outside the roots:
\[ y \leq 5 \quad \text{or} \quad y \geq 9 \]

### Final Step: State the range
Thus, the range of the expression is:
\[ (-\infty, 5] \cup [9, \infty) \]