b) [tex]$a^2-10 a+16-6 b-b^2$[/tex]
Answer (1)
Complete the square for the terms involving a : a 2 − 10 a = ( a − 5 ) 2 − 25 .
Complete the square for the terms involving b : − b 2 − 6 b = − ( b + 3 ) 2 + 9 .
Substitute these back into the original expression and simplify: ( a − 5 ) 2 − 25 + 16 − 6 b − b 2 = ( a − 5 ) 2 − ( b + 3 ) 2 .
Factor the difference of squares: ( a − 5 ) 2 − ( b + 3 ) 2 = ( a + b − 2 ) ( a − b − 8 ) .
The final factored expression is ( a + b − 2 ) ( a − b − 8 ) .
Explanation
Understanding the Problem We are given the expression a 2 − 10 a + 16 − 6 b − b 2 and we want to rewrite it by completing the square and factoring.
Completing the Square for a First, let's complete the square for the terms involving a . We have a 2 − 10 a . To complete the square, we need to add and subtract ( 2 10 ) 2 = 5 2 = 25 . So, we can rewrite a 2 − 10 a as a 2 − 10 a + 25 − 25 = ( a − 5 ) 2 − 25 .
Substituting Back Now, substitute this back into the original expression: ( a − 5 ) 2 − 25 + 16 − 6 b − b 2 = ( a − 5 ) 2 − 9 − 6 b − b 2 .
Completing the Square for b Next, let's complete the square for the terms involving b . We have − b 2 − 6 b . We can rewrite this as − ( b 2 + 6 b ) . To complete the square, we need to add and subtract ( 2 6 ) 2 = 3 2 = 9 inside the parenthesis. So, we have − ( b 2 + 6 b + 9 − 9 ) = − ( b + 3 ) 2 + 9 .
Substituting Back Again Substitute this back into the expression: ( a − 5 ) 2 − 9 − ( b 2 + 6 b ) = ( a − 5 ) 2 − 9 − ( b 2 + 6 b + 9 − 9 ) = ( a − 5 ) 2 − 9 − ( b + 3 ) 2 + 9 = ( a − 5 ) 2 − ( b + 3 ) 2 .
Factoring the Difference of Squares Now we have a difference of squares: ( a − 5 ) 2 − ( b + 3 ) 2 . We can factor this as ( a − 5 + b + 3 ) ( a − 5 − b − 3 ) = ( a + b − 2 ) ( a − b − 8 ) .
Final Answer Therefore, the factored expression is ( a + b − 2 ) ( a − b − 8 ) .
Examples
Factoring expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Similarly, economists use factoring to analyze market trends and predict future economic behavior. By understanding how to factor expressions, you can solve a wide range of problems in various fields.
