Question

Factor the expression.
81 x squared plus 72 x plus 1681x2+72x+16

Answer 1

To factor the expression \( 81x^2 + 72x + 1681 \), we can first check if it can be factored as a quadratic expression.

The general form of a quadratic expression is \( ax^2 + bx + c \). Here, we have:

- \( a = 81 \)
- \( b = 72 \)
- \( c = 1681 \)

To factor this expression, we can use the method of finding two numbers that multiply to \( ac = 81 \times 1681 \) and add up to \( b = 72 \).

Calculating \( ac \):
\[
81 \times 1681 = 136761
\]

Now, we need two numbers that multiply to \( 136761 \) and add up to \( 72 \). This can be difficult without a calculator, so let's proceed with an alternate method by trying different kinds of factorizations, or by completing the square or using the quadratic formula to find roots.

Given the coefficients are large, it might be better to check for perfect squares or common factors.

Next, we can try solving the quadratic equation by using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Calculating the discriminant \( b^2 - 4ac \):
\[
b^2 = 72^2 = 5184
\]
\[
4ac = 4 \cdot 81 \cdot 1681 = 544920
\]
\[
b^2 - 4ac = 5184 - 544920 = -539736
\]

Since the discriminant is negative, the expression \( 81x^2 + 72x + 1681 \) does not factor over the real numbers and has complex roots.

Thus, this expression cannot be factored using real number factors. If you require complex factorization or have further context like a specific domain of numbers, let me know!