To determine if the quadratic expression [tex]\(1 + 14g + 49g^2\)[/tex] is a perfect square, we need to see if it can be written in the form [tex]\((ag + b)^2\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants.
1. Original Expression:
[tex]\[
1 + 14g + 49g^2
\][/tex]
2. General Form of a Perfect Square:
[tex]\((ag + b)^2 = a^2g^2 + 2abg + b^2\)[/tex]
By comparing both expressions, we can identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a^2g^2\)[/tex] should match [tex]\(49g^2\)[/tex]
- [tex]\(2abg\)[/tex] should match [tex]\(14g\)[/tex]
- [tex]\(b^2\)[/tex] should match the constant term [tex]\(1\)[/tex]
3. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
From [tex]\(49g^2\)[/tex], we have:
[tex]\[
a^2 = 49 \implies a = 7 \text{ (since } 7^2 = 49\text{)}
\][/tex]
From [tex]\(14g\)[/tex], we have:
[tex]\[
2ab = 14 \implies 2(7)b = 14 \implies 14b = 14 \implies b = 1
\][/tex]
From [tex]\(1\)[/tex], we have:
[tex]\[
b^2 = 1 \implies b = 1 \text{ (since } 1^2 = 1\text{)}
\][/tex]
4. Construct the Perfect Square Form:
Now that we have [tex]\(a = 7\)[/tex] and [tex]\(b = 1\)[/tex], we can write the factored form of the expression:
[tex]\[
7g + 1
\][/tex]
Therefore, the perfect square of the expression is:
[tex]\[
(7g + 1)^2
\][/tex]
Thus, the factorization of [tex]\(1 + 14g + 49g^2\)[/tex] is [tex]\((7g + 1)^2\)[/tex].
