To factor the perfect square trinomials into the required forms, we can use the general formulas:
- \( a^2 + 2ab + b^2 = (a + b)^2 \)
- \( a^2 - 2ab + b^2 = (a - b)^2 \)
Let's factor each of the given trinomials:
-
\( x^2 + 6x + 9 \)
Here, \( a = x \) and \( b = 3 \) (since \( b^2 = 9 \)).
So, \( x^2 + 6x + 9 = (x + 3)^2 \)
-
\( x^2 + 10x + 25 \)
Here, \( a = x \) and \( b = 5 \) (since \( b^2 = 25 \)).
So, \( x^2 + 10x + 25 = (x + 5)^2 \)
-
\( 9x^2 + 12x + 4 \)
Here, \( a = 3x \) and \( b = 2 \) (since \( b^2 = 4 \)).
So, \( 9x^2 + 12x + 4 = (3x + 2)^2 \)
-
\( 9x^2 - 30x + 25 \)
Here, \( a = 3x \) and \( b = 5 \) (since \( b^2 = 25 \)).
So, \( 9x^2 - 30x + 25 = (3x - 5)^2 \)
Now, summarizing the factorizations:
-
\( x^2 + 6x + 9 = (x + 3)^2 \)
-
\( x^2 + 10x + 25 = (x + 5)^2 \)
-
\( 9x^2 + 12x + 4 = (3x + 2)^2 \)
-
\( 9x^2 - 30x + 25 = (3x - 5)^2 \)