Problem 5 - Algebra A Combined Function

Question

Find the value of the following expressions when \(x=2, y=-2,\) and \(z=-5\). $$ \frac{x^{2}+8 x+2}{x^{2}-x-6} $$

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{11}{2}\).

Step 1: Substitute the given values into the expression

Start by substituting the values \(x = 2\), \(y = -2\), and \(z = -5\) into each instance of \(x\), \(y\), and \(z\) in the expression: \[ \frac{x^{2} + 8x + 2}{x^{2} - x - 6} \] becomes \[ \frac{2^{2} + 8(2) + 2}{2^{2} - 2 - 6}. \]

Step 2: Calculate the numerator

Calculate the value of the numerator by evaluating \(2^{2} + 8(2) + 2\). Begin by computing \(2^{2} = 4\), then \(8 \times 2 = 16\). Add these results to get \(4 + 16 = 20\). Finally, add the constant \(2\) to get \(20 + 2 = 22\).

Step 3: Calculate the denominator

Calculate the value of the denominator by evaluating \(2^{2} - 2 - 6\). First, compute \(2^{2} = 4\). Then subtract \(2\) to get \(4 - 2 = 2\). Finally, subtract \(6\) which results in \(2 - 6 = -4\).

Step 4: Divide the numerator by the denominator

Now divide the numerator by the denominator: \(\frac{22}{-4}\). Simplifying this fraction, divide both numerator and denominator by 2 to get \(\frac{22 \div 2}{-4 \div 2} = \frac{11}{-2}\). Therefore, the value of the expression is \(-\frac{11}{2}\).

Key Concepts

Substitution Method

To solve algebraic expressions by substitution, we replace the variables with their given values. In our example, we need to substitute the values for \( x \), \( y \), and \( z \) into the expression, though only \( x \) is relevant here since it is the only variable in the expression.

We begin with the original expression:

• \( \frac{x^{2} + 8x + 2}{x^{2} - x - 6} \)

Substitute \( x = 2 \) into every instance of \( x \) in the expression:

• \( \frac{2^{2} + 8(2) + 2}{2^{2} - 2 - 6} \)

By substituting, the problem becomes an arithmetic calculation, making it much simpler to solve. This method helps to evaluate the expression with the specific condition provided.

Numerator and Denominator Calculation

When dealing with fractions, it's crucial to calculate both the numerator and the denominator correctly. Let's break it down into steps.

Numerator Calculation:

• Start with \(2^{2} + 8(2) + 2\)
• Calculate \(2^{2} = 4\)
• Then \(8 \times 2 = 16\)
• Adding these gives \(4 + 16 = 20\)
• Finally, add the constant 2 to obtain \(20 + 2 = 22\)

Denominator Calculation:

• First, compute \(2^{2} = 4\)
• Subtract 2: \(4 - 2 = 2\)
• Then subtract 6: \(2 - 6 = -4\)

Understanding these calculations is key to simplifying and comparing fractions correctly.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form. This involves finding the greatest common divisor (GCD) and dividing both the numerator and the denominator by it.

In our example, the fraction is \( \frac{22}{-4} \). To simplify:

• Find the GCD of 22 and 4, which is 2.
• Divide the numerator by 2: \(22 \div 2 = 11\)
• Divide the denominator by 2: \(-4 \div 2 = -2\)
• The simplified fraction is then \(\frac{11}{-2}\) or \(-\frac{11}{2}\)

It's important to express fractions in their simplest form to make them easier to understand and compare.