To determine the last term required to make a given expression a perfect square trinomial, we use the structure of a perfect square trinomial:
For a general perfect square trinomial of the form [tex]\(a^2 + 2ab + b^2\)[/tex], we observe that the last term ([tex]\(b^2\)[/tex]) is the square of half the coefficient of the middle term.
Let's analyze each of the given trinomials:
1. [tex]\(x^2 + 12x + 36 = (x + 6)^2\)[/tex]
In this case, the expression is already given as a perfect square trinomial. The last term is 36, confirming that it is [tex]\(6^2\)[/tex], where 6 is half of the middle term's coefficient (12).
2. [tex]\(x^2 - 18x + 81 = (x - 9)^2\)[/tex]
Here, the expression shows that the last term is 81, which is [tex]\(9^2\)[/tex], where 9 is half of the middle term's coefficient (18).
3. [tex]\(x^2 + 5x + \underline{\phantom{0}}\)[/tex]
To make [tex]\(x^2 + 5x\)[/tex] a perfect square trinomial, we need to find the last term.
- Take the coefficient of the middle term (5).
- Divide it by 2: [tex]\(\frac{5}{2} = 2.5\)[/tex].
- Square it: [tex]\(2.5^2 = 6.25\)[/tex].
Therefore, to complete the square, the last term should be 6.25.
So, the perfect square trinomial form will be:
[tex]\[x^2 + 5x + 6.25 = \left(x + \frac{5}{2}\right)^2 = (x + 2.5)^2\][/tex]
4. [tex]\(4x^2 + 12x + 9 = (2x + 3)^2\)[/tex]
This expression is already in the form of a perfect square trinomial. The last term is [tex]\(3^2 = 9\)[/tex], where 3 is half of the middle term's coefficient (12), adjusted for the leading coefficient of 4 (since [tex]\((2x)^2 = 4x^2\)[/tex]).
In summary, the trinomials are:
1. [tex]\(x^2 + 12x + 36 = (x + 6)^2\)[/tex]
2. [tex]\(x^2 - 18x + 81 = (x - 9)^2\)[/tex]
3. [tex]\(x^2 + 5x + 6.25 = (x + 2.5)^2\)[/tex]
4. [tex]\(4x^2 + 12x + 9 = (2x + 3)^2\)[/tex]
