Certainly! Let's convert each expression into its equivalent square form.
### Part a)
Expression: [tex]\(x^2 + 6x + 9\)[/tex]
Let's find the square form:
[tex]\[
x^2 + 6x + 9 = (x + 3)^2
\][/tex]
### Part b)
Expression: [tex]\(x^2 - 8x + 16\)[/tex]
Let's find the square form:
[tex]\[
x^2 - 8x + 16 = (x - 4)^2
\][/tex]
### Part c)
Expression: [tex]\(4a^2 + 4ab + b^2\)[/tex]
Let's find the square form:
[tex]\[
4a^2 + 4ab + b^2 = (2a + b)^2
\][/tex]
### Part d)
Expression: [tex]\(p^2 - 6pq + 9q^2\)[/tex]
Let's find the square form:
[tex]\[
p^2 - 6pq + 9q^2 = (p - 3q)^2
\][/tex]
### Part e)
Expression: [tex]\(4x^2 + 12xy + 9y^2\)[/tex]
Let's find the square form:
[tex]\[
4x^2 + 12xy + 9y^2 = (2x + 3y)^2
\][/tex]
### Part f)
Expression: [tex]\(25x^2 - 40xy + 16y^2\)[/tex]
Let's find the square form:
[tex]\[
25x^2 - 40xy + 16y^2 = (5x - 4y)^2
\][/tex]
### Part g)
Expression: [tex]\(49a^2 - 42ab + 9b^2\)[/tex]
Let's find the square form:
[tex]\[
49a^2 - 42ab + 9b^2 = (7a - 3b)^2
\][/tex]
### Part h)
Expression: [tex]\(x^2 + 2 + \frac{1}{x^2}\)[/tex]
Let's find the square form:
[tex]\[
x^2 + 2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2
\][/tex]
### Part i)
Expression: [tex]\(4p^2 - 2 + \frac{1}{4p^2}\)[/tex]
Let's find the square form:
[tex]\[
4p^2 - 2 + \frac{1}{4p^2} = \left(2p - \frac{1}{2p}\right)^2 - 2
\][/tex]
By expressing the provided polynomials in square forms, we've simplified them to make it easier to identify their structure and potential solutions.
