Write the trinomial as a square of a binomial or as an expression opposite to a square of a binomial:
–44ax +121a^2+4x^2
rearrange to standard form, to make it easier to check
121a^2 - 44ax + 4x^2
Note that the two end terms are perfect squares: (11a)^2 and (2x)^2
Now see that 44 = 2(11*2)
So, it is clearly
(11a-2x)^2
or (2x-11a)^2
To write the trinomial –44ax + 121a^2 + 4x^2 as a square of a binomial or as an expression opposite to a square of a binomial, we need to identify perfect square terms.
The first term –44ax does not have any perfect square terms.
The second term 121a^2 is a perfect square. The square root of 121a^2 is 11a. Therefore, we can rewrite 121a^2 as (11a)^2.
The third term 4x^2 is also a perfect square. The square root of 4x^2 is 2x. Hence, we can rewrite 4x^2 as (2x)^2.
Now we can rewrite the trinomial as:
(11a)^2 - 2(11a)(2x) + (2x)^2
Simplifying this further gives us:
(11a - 2x)^2
To write the given trinomial as a square of a binomial or as an expression opposite to a square of a binomial, we can look for patterns that match the square of a binomial form:
(a ± b)^2 = a^2 ± 2ab + b^2
Looking at the given trinomial: -44ax + 121a^2 + 4x^2, we notice that the first term is a product of -4, 4, and a, which is similar to (2a)^2. Similarly, the last term is a perfect square of x, which is similar to (2x)^2.
To express the middle term, we can take twice the product of the square root of the first and last term: (2a)(2x) = 4ax.
Now, let's rewrite the trinomial using these patterns:
-44ax + 121a^2 + 4x^2
= (11a)^2 - 2(11a)(2x) + (2x)^2
= (11a - 2x)^2
Therefore, the given trinomial can be written as the square of a binomial: (11a - 2x)^2.