Question
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Answer
The expression factors to \( (9x + 1)^2 \).
Solution
1. Recognize that the quadratic is in the form \( 81x^2 + 18x + 1 \) and that \(81 = 9^2\) and \(1 = 1^2\).
2. Compare the middle term with \(2 \times 9x \times 1\):
\[
2 \times 9x \times 1 = 18x
\]
Since it matches the middle term, the quadratic is a perfect square.
3. Therefore, the factorization is:
\[
(9x + 1)^2
\]
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Simplify this solution Beyond the Answer
To factor the expression \( 81x^{2} + 18x + 1 \), we can recognize that this can be rewritten in a perfect square form. Notice that \( 81x^{2} \) is \( (9x)^{2} \), \( 18x \) is \( 2 \cdot 9x \cdot 1 \), and \( 1 \) is \( 1^{2} \). Thus, we have: \[ 81x^{2} + 18x + 1 = (9x + 1)^{2} \] So the factored form is: \[ (9x + 1)^{2} \] This concise expression beautifully encapsulates the quadratic's essence in a perfect square format!
