Solving Quadratic Equations and Evaluating Expressions

In this article, we will delve into the process of solving a specific quadratic equation and evaluating complex algebraic expressions, providing a detailed step-by-step solution. This content is designed for students and professionals interested in mathematics and problem-solving techniques.

Solving the Quadratic Equation x^2 - 12x - 33 0

Given the quadratic equation x^2 - 12x - 33 0, our first step is to solve for x. To do this, we will use the quadratic formula:

?????? x -b ± sqrt{b^2 - 4ac}}{2a}

where a 1, b -12, and c -33. Let's proceed with the calculations.

Step 1: Calculating the Discriminant

The discriminant is given by:

b^2 - 4ac (-12)^2 - 4(1)(-33) 144 132 276

Step 2: Finding the Roots

Using the quadratic formula:

x -(-12) ± sqrt{276}}{2(1)} 12 ± sqrt{276}

This simplifies to:

x 12 ± 2sqrt{69}

Step 3: Finding the Roots in the Form of x - 4

Now, let's find x - 4 for each root:

For x_1 12 2sqrt{69}, x_1 - 4 8 2sqrt{69}

For x_2 12 - 2sqrt{69}, x_2 - 4 8 - 2sqrt{69}

Evaluating the Expression x - 4^2 cdot frac{1}{x - 4^2}

Step 4: Calculating x - 4^2

First, let's calculate x - 4^2

For x_1 8 2sqrt{69}, x_1 - 4^2 (8 2sqrt{69})^2 64 32sqrt{69} 276 340 32sqrt{69}

For x_2 8 - 2sqrt{69}, x_2 - 4^2 (8 - 2sqrt{69})^2 64 - 32sqrt{69} 276 340 - 32sqrt{69}

Step 5: Dividing by x - 4^2

Next, we need to find frac{1}{x - 4^2}

For x_1 - 4^2 340 32sqrt{69}, frac{1}{340 32sqrt{69}} can be simplified by multiplying by the conjugate:

frac{1}{340 32sqrt{69}} cdot frac{340 - 32sqrt{69}}{340 - 32sqrt{69}} frac{340 - 32sqrt{69}}{340^2 - (32sqrt{69})^2} frac{340 - 32sqrt{69}}{115600 - 62496} frac{340 - 32sqrt{69}}{53104} frac{85 - 8sqrt{69}}{13276}

Similarly, for x_2 - 4^2 340 - 32sqrt{69}, frac{1}{340 - 32sqrt{69}} cdot frac{340 32sqrt{69}}{340 32sqrt{69}} frac{340 32sqrt{69}}{115600 - 62496} frac{340 32sqrt{69}}{53104} frac{85 8sqrt{69}}{13276}

Step 6: Combining the Results

Finally, we need to compute both expressions:

1. For x_1 - 4^2 340 32sqrt{69}, the expression is:

(340 32sqrt{69}) cdot frac{85 - 8sqrt{69}}{13276} frac{29400 2704sqrt{69} cdot 85 - 8(32)69} 14

2. For x_2 - 4^2 340 - 32sqrt{69}, the expression is:

(340 - 32sqrt{69}) cdot frac{85 8sqrt{69}}{13276} frac{29400 2704sqrt{69} cdot 85} 14

Therefore, in both cases, the final answer is:

boxed{14}