Find the value of the expression when [tex]x=4[/tex].
[tex]\frac{x^2+4 x+8}{x^2-8 x-8}=[/tex]
(Type an integer or a simplified fraction.)
Answer (1)
Substitute x = 4 into the expression x 2 − 8 x − 8 x 2 + 4 x + 8 .
Calculate the numerator: 4 2 + 4 ( 4 ) + 8 = 40 .
Calculate the denominator: 4 2 − 8 ( 4 ) − 8 = − 24 .
Simplify the fraction: − 24 40 = − 3 5 .
− 3 5
Explanation
Understanding the problem We are given the expression x 2 − 8 x − 8 x 2 + 4 x + 8 and asked to evaluate it at x = 4 .
Substituting the value of x First, we substitute x = 4 into the expression: 4 2 − 8 ( 4 ) − 8 4 2 + 4 ( 4 ) + 8
Simplifying the numerator Next, we simplify the numerator: 4 2 + 4 ( 4 ) + 8 = 16 + 16 + 8 = 40
Simplifying the denominator Then, we simplify the denominator: 4 2 − 8 ( 4 ) − 8 = 16 − 32 − 8 = − 24
Simplifying the fraction Now, we have the fraction − 24 40 . We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 8: − 24 40 = − 24 ÷ 8 40 ÷ 8 = − 3 5 = − 3 5
Final Answer Therefore, the value of the expression when x = 4 is − 3 5 .
Examples
Evaluating algebraic expressions is a fundamental skill in mathematics with numerous real-world applications. For instance, engineers use such evaluations to determine the stress on a structural component under specific loads. Suppose the expression represents the stress on a beam, where 'x' is a parameter related to the applied force. By substituting a particular value for 'x', engineers can quickly calculate the stress and ensure it remains within safe limits, preventing potential failures. This ensures safety and efficiency in design and construction.
