Factorise [tex]$5 a^2-10 a+5$
Answer (2)
The expression 5 a 2 − 10 a + 5 can be factored by first taking out the common factor of 5, resulting in 5 ( a 2 − 2 a + 1 ) . The quadratic part a 2 − 2 a + 1 is a perfect square trinomial, which factors to ( a − 1 ) 2 . Therefore, the final factorization is 5 ( a − 1 ) 2 .
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Factor the expression 12.10 x as 12.1 x .
Factor out the common factor 5 from 5 a 2 − 10 a + 5 , resulting in 5 ( a 2 − 2 a + 1 ) .
Recognize a 2 − 2 a + 1 as a perfect square trinomial and factor it as ( a − 1 ) 2 .
The complete factorization of 5 a 2 − 10 a + 5 is 5 ( a − 1 ) 2 , so the final answers are 12.1 x and 5 ( a − 1 ) 2 .
Explanation
Problem Analysis We are asked to factorize two expressions:
12.10 x
5 a 2 − 10 a + 5
Factoring the first expression Let's factorize the first expression, 12.10 x . This expression can be written as 12.1 × x . Since 12.1 is already a decimal number, we can rewrite it as a fraction to express it in its simplest form. 12.1 = 10 121 . Therefore, the expression can be written as 10 121 x . There are no common factors to further simplify, so the factored form remains as 12.1 x or 10 121 x .
Factoring the second expression Now, let's factorize the second expression, 5 a 2 − 10 a + 5 . We can first factor out the common factor of 5 from each term: 5 ( a 2 − 2 a + 1 ) . Now, we need to factorize the quadratic expression a 2 − 2 a + 1 . This is a perfect square trinomial, which can be factored as ( a − 1 ) 2 . Therefore, the complete factorization of the second expression is 5 ( a − 1 ) 2 .
Final Answer In summary:
The factorization of 12.10 x is 12.1 x .
The factorization of 5 a 2 − 10 a + 5 is 5 ( a − 1 ) 2 .
Examples
Factoring is a fundamental concept in algebra and is used in various real-life applications. For instance, when designing a rectangular garden with an area represented by a quadratic expression, factoring helps determine the possible dimensions of the garden. Similarly, in physics, factoring can simplify equations describing projectile motion, making it easier to calculate the time it takes for an object to reach a certain height. Understanding factoring allows for efficient problem-solving and optimization in many practical scenarios.
