Factor completely.
$5 x^2-45$
A. $5(x+3)(x-3)$
B. $5\left(x^2-45\right)$
C. $(5 x+9)(x-5)$
D. $5(x-9)(x+9)$
Answer (2)
The expression 5 x 2 − 45 can be factored completely as 5 ( x + 3 ) ( x − 3 ) by first factoring out the greatest common factor of 5 and then applying the difference of squares formula. Therefore, the correct option is A. 5 ( x + 3 ) ( x − 3 ) .
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Factor out the greatest common factor: 5 x 2 − 45 = 5 ( x 2 − 9 ) .
Recognize the difference of squares: x 2 − 9 = ( x + 3 ) ( x − 3 ) .
Combine the factors: 5 ( x 2 − 9 ) = 5 ( x + 3 ) ( x − 3 ) .
The completely factored expression is 5 ( x + 3 ) ( x − 3 ) .
Explanation
Understanding the Problem We are asked to factor the expression 5 x 2 − 45 completely. This means we want to break it down into its simplest factors.
Factoring out the GCF First, we look for the greatest common factor (GCF) of the terms 5 x 2 and − 45 . The GCF is 5. We factor out the 5: 5 x 2 − 45 = 5 ( x 2 − 9 )
Factoring the Difference of Squares Now, we examine the expression inside the parentheses: x 2 − 9 . This is a difference of squares, since x 2 is a perfect square and 9 = 3 2 is also a perfect square. We can use the difference of squares factorization: a 2 − b 2 = ( a + b ) ( a − b ) . In our case, a = x and b = 3 . Therefore, we have: x 2 − 9 = ( x + 3 ) ( x − 3 )
Final Factored Expression Substituting this back into our expression, we get: 5 ( x 2 − 9 ) = 5 ( x + 3 ) ( x − 3 ) So, the completely factored expression is 5 ( x + 3 ) ( x − 3 ) .
Examples
Factoring 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, ensuring stability and efficiency. Similarly, economists use factoring to analyze market trends and predict economic behavior. By breaking down complex problems into simpler components, factoring allows professionals to make informed decisions and optimize outcomes.
