Factorise [tex]$5 a^2-10 a+5$

The factorized form of the expression 5 a 2 − 10 a + 5 is 5 ( a − 1 ) 2 . This is achieved by first factoring out the common factor of 5 and then recognizing the quadratic expression as a perfect square trinomial. The final result reflects these two steps clearly and concisely.
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Answered by Anonymous | 2025-07-04

Factor out the common factor 5: 5 a 2 − 10 a + 5 = 5 ( a 2 − 2 a + 1 ) .
Recognize the quadratic expression as a perfect square trinomial: a 2 − 2 a + 1 = ( a − 1 ) 2 .
Substitute back into the expression: 5 ( a 2 − 2 a + 1 ) = 5 ( a − 1 ) 2 .
The final factorized form is: 5 ( a − 1 ) 2 ​ .

Explanation

Understanding the Problem We are asked to factorize the expression 5 a 2 − 10 a + 5 . This looks like a quadratic expression, and our goal is to rewrite it as a product of simpler terms.

Factoring out the Common Factor First, notice that each term in the expression has a common factor of 5. We can factor this out to simplify the expression: 5 a 2 − 10 a + 5 = 5 ( a 2 − 2 a + 1 ) Now we need to factor the quadratic expression inside the parentheses: a 2 − 2 a + 1 .

Recognizing the Perfect Square Trinomial We can recognize the expression a 2 − 2 a + 1 as a perfect square trinomial. A perfect square trinomial is a trinomial that can be written as the square of a binomial. In this case, we have: a 2 − 2 a + 1 = ( a − 1 ) 2 This is because ( a − 1 ) 2 = ( a − 1 ) ( a − 1 ) = a 2 − a − a + 1 = a 2 − 2 a + 1 .

Writing the Final Factorized Form Now, we substitute this back into our expression: 5 ( a 2 − 2 a + 1 ) = 5 ( a − 1 ) 2 So, the fully factorized form of the expression is 5 ( a − 1 ) 2 .

Final Answer Therefore, the factorized form of the given expression 5 a 2 − 10 a + 5 is 5 ( a − 1 ) 2 .


Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you want to find the dimensions of a rectangular garden with an area represented by the expression 5 a 2 − 10 a + 5 , where 'a' is a variable related to the dimensions. By factoring the expression to 5 ( a − 1 ) 2 , you can determine that the garden is a square with side length 5 ​ ( a − 1 ) . This skill is also crucial in physics for solving problems involving projectile motion or energy calculations, where quadratic equations often arise.

Answered by GinnyAnswer | 2025-07-04