Multiply.
[tex](5 x-1)(5 x-1)[/tex]
Answer (2)
Multiply the first terms: ( 5 x ) ( 5 x ) = 25 x 2 .
Multiply the outer and inner terms: ( 5 x ) ( − 1 ) + ( − 1 ) ( 5 x ) = − 10 x .
Multiply the last terms: ( − 1 ) ( − 1 ) = 1 .
Combine the terms to get the final answer: 25 x 2 − 10 x + 1
Explanation
Understanding the Problem We are asked to multiply the expression ( 5 x − 1 ) ( 5 x − 1 ) and choose the correct answer from the given options. This is a straightforward application of the distributive property (often remembered by the acronym FOIL, which stands for First, Outer, Inner, Last).
Applying the Distributive Property To multiply ( 5 x − 1 ) ( 5 x − 1 ) , we apply the distributive property (FOIL method):
First: ( 5 x ) ( 5 x ) = 25 x 2 Outer: ( 5 x ) ( − 1 ) = − 5 x Inner: ( − 1 ) ( 5 x ) = − 5 x Last: ( − 1 ) ( − 1 ) = 1
So, ( 5 x − 1 ) ( 5 x − 1 ) = 25 x 2 − 5 x − 5 x + 1 .
Simplifying the Expression Now, we simplify the expression by combining like terms:
25 x 2 − 5 x − 5 x + 1 = 25 x 2 − 10 x + 1
Choosing the Correct Answer Comparing our result, 25 x 2 − 10 x + 1 , with the given options, we find that it matches the second option.
Therefore, the correct answer is 25 x 2 − 10 x + 1 .
Examples
Understanding how to expand and simplify algebraic expressions like ( 5 x − 1 ) ( 5 x − 1 ) is fundamental in many areas, such as physics, engineering, and economics. For instance, when calculating the area of a square with side length ( 5 x − 1 ) , you would need to expand this expression. Similarly, in physics, you might encounter such expressions when dealing with quadratic equations describing projectile motion or energy calculations. Mastering these algebraic manipulations allows for accurate modeling and problem-solving in various real-world scenarios.
The expression ( 5 x − 1 ) ( 5 x − 1 ) multiplies to give 25 x 2 − 10 x + 1 using the FOIL method. We calculate the first, outer, inner, and last terms, and then simplify by combining like terms. The final answer is 25 x 2 − 10 x + 1 .
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