Problem 13 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(x=6 .\) See Example 1. $$ \frac{x^{2}-4 x-12}{x^{2}+x-2} $$

Step-by-Step Solution

Answer

The value of the expression for \(x=6\) is 0.

Step 1: Substitute the Value of x

Given the expression, first substitute the value of \(x = 6\). So the expression \( \frac{x^{2}-4x-12}{x^{2}+x-2} \) becomes \( \frac{6^{2}-4(6)-12}{6^{2}+6-2} \).

Step 2: Simplify the Numerator

Simplify the numerator by calculating each term: \(6^{2} = 36\), \(-4(6) = -24\), and then subtract 12. Therefore, the numerator is \(36 - 24 - 12 = 0\).

Step 3: Simplify the Denominator

Simplify the denominator by calculating each term: \(6^{2} = 36\), add 6, and subtract 2. Therefore, the denominator is \(36 + 6 - 2 = 40\).

Step 4: Evaluate the Fraction

Substitute the simplified numerator and denominator back into the fraction. Since the numerator is 0, the entire expression simplifies to \(\frac{0}{40} = 0\).

Key Concepts

Expression Evaluation

Expression evaluation involves finding the value of an algebraic expression after substituting numerical values for its variables. In our exercise, we have an algebraic fraction \[ \frac{x^{2}-4x-12}{x^{2}+x-2}. \]To evaluate this expression for a specific value of \(x\), in this case, \(x=6\), follow these steps:

• Substitute the given value into the expression.
• Perform arithmetic operations carefully.
• Ensure that each operation follows the correct order: powers, multiplication/division, and then addition/subtraction.

By plugging \(x=6\) into the fraction, the expression changes, setting the stage for further calculations and simplification.

Substitution Method

The substitution method is a technique used to replace variables with their given numerical values in an expression. This process makes abstract algebraic expressions concrete and manageable.Here’s how it works using our exercise. Substitute \(x=6\) directly into the expression:\[\frac{6^{2} - 4(6) - 12}{6^{2} + 6 - 2}.\]Remember:

• Carefully replace each occurrence of \(x\) in the expression with the value 6.
• Pay attention to signs and operations while substituting, to prevent errors.
• After substitution, proceed to simplify each part (numerator and denominator separately).

This approach simplifies the evaluation of the given expression by transforming it into a numerical form.

Simplification Process

Once you've substituted a numerical value into an algebraic expression, the next step is simplification. Simplification helps in reducing an expression to its simplest form, making it easier to interpret the result.In our exercise:**Simplifying the Numerator**- Calculate \(6^{2} = 36\) then subtract \(-4 \times 6 = -24\) and finally subtract \(12\).- This results in \(36 - 24 - 12 = 0\).**Simplifying the Denominator**- Calculate \(6^{2} = 36\) then add \(6\) and subtract \(2\).- The result is \(36 + 6 - 2 = 40\).**Final Expression**- Substitute these simplified results back into the fraction \(\frac{0}{40}\), which yields \(0\).Simplification ensures that you arrive at a neat and easily interpretable answer, avoiding mistakes and ensuring consistency in your calculations.