To determine whether each given expression is a perfect square trinomial (PST), we need to check if the quadratic expression can be written in the form [tex]\((ax + b)^2 = a^2x^2 + 2abx + b^2\)[/tex].
Let's analyze each expression step-by-step.
### a. [tex]\(x^2 + 6x + 9\)[/tex]
1. Identify the coefficients: [tex]\(a = 1\)[/tex] (coefficient of [tex]\(x^2\)[/tex]), [tex]\(b = 6\)[/tex] (coefficient of [tex]\(x\)[/tex]), and [tex]\(c = 9\)[/tex] (constant term).
2. Check if [tex]\(c = b^2 / (4a)\)[/tex]:
[tex]\[
c = \left(\frac{6}{2}\right)^2 = 3^2 = 9
\][/tex]
Since 9 equals the constant term, the expression is a perfect square trinomial.
3. Factor the trinomial:
[tex]\[
x^2 + 6x + 9 = (x + 3)^2
\][/tex]
Thus, [tex]\(x^2 + 6x + 9\)[/tex] is a PST.
### b. [tex]\(x^2 - 10x - 25\)[/tex]
1. Identify the coefficients: [tex]\(a = 1\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -25\)[/tex].
2. Check if [tex]\(c = b^2 / (4a)\)[/tex]:
[tex]\[
c = \left(\frac{-10}{2}\right)^2 = (-5)^2 = 25
\][/tex]
Here, 25 does not equal the constant term [tex]\(c = -25\)[/tex].
Thus, [tex]\(x^2 - 10x - 25\)[/tex] is not a PST.
### c. [tex]\(9x^2 - 12x + 9\)[/tex]
1. Identify the coefficients: [tex]\(a = 9\)[/tex], [tex]\(b = -12\)[/tex], and [tex]\(c = 9\)[/tex].
2. Check if [tex]\(c = b^2 / (4a)\)[/tex]:
[tex]\[
c = \frac{(-12)^2}{4 \cdot 9} = \frac{144}{36} = 4
\][/tex]
Here, 4 does not equal the constant term [tex]\(c = 9\)[/tex].
Thus, [tex]\(9x^2 - 12x + 9\)[/tex] is not a PST.
### d. [tex]\(4x^2 + 28xy + 49y^2\)[/tex]
1. Identify the structure: [tex]\(a = 2x\)[/tex], [tex]\(b = 7y\)[/tex].
2. Rewrite the expression:
[tex]\[
(2x)^2 + 2(2x)(7y) + (7y)^2 = 4x^2 + 28xy + 49y^2
\][/tex]
Thus, [tex]\(4x^2 + 28xy + 49y^2\)[/tex] is a PST and can be factored as:
[tex]\[
(2x + 7y)^2
\][/tex]
In summary:
a. [tex]\(x^2 + 6x + 9\)[/tex] is a PST.
b. [tex]\(x^2 - 10x - 25\)[/tex] is NOT a PST.
c. [tex]\(9x^2 - 12x + 9\)[/tex] is NOT a PST.
d. [tex]\(4x^2 + 28xy + 49y^2\)[/tex] is a PST.
