Question

How do you factor \[3{n^2} - 30n + 75\] the trinomial completely?

Answer 1

Hint: The question is to factorize the given expression. In Mathematics, factorization means separating or breaking off an expression or entity into a product including other expressions or factors, in which when they are multiplied give the original number.
Formula used: \[{(a \pm b)^2} = {a^2} + {b^2} \pm 2ab\]

Complete step by step solution:
The given expression is:
 \[3{n^2} - 30n + 75\]
As, we can see that the number 3 is common in all the terms, so we will take out 3 as a common factor from the expression, and we get:
 \[ = 3({n^2} - 10n + 25)\]
We will try the basic formula of algebra that is:
 \[{(a \pm b)^2} = {a^2} + {b^2} \pm 2ab\]
Now, we know that this is a perfect square. According to the formula, if we see our given expression, we can see that \[a = n\] and \[b = - 5\] . Here, \[ \Rightarrow 2ab = 2 \times n \times ( - 5) = 10n\] .
Now, we can see that \[{n^2} - 10n + 25\] is also a perfect square according to the above given formula. So, if we try to solve the given expression according to the formula, then we get:
 \[ = {(n - 5)^2}\]
Hence, \[ \Rightarrow 3({n^2} - 10n + 25) = 3{(n - 5)^2}\] .
Therefore, we got our result as \[3{(n - 5)^2}\] .
So, the correct answer is “ \[3{(n - 5)^2}\] ”.

Note: Any expression that is obtained from the square of a binomial equation is called a perfect square trinomial. The rule or condition of any expression to be a perfect square trinomial is that it should be in the form of \[a{x^2} + bx + c\] and it should also satisfy the condition that is \[{b^2} = 4ac\] .
The above expression is a trinomial expression. Trinomial expression means that an expression is having three terms, or we can say that any polynomial has three terms. The above formula in the solution is a perfect square trinomial rule.