Question

Find the area of the square. Simplify your answer.
the square equation is 3x^2 - 5

Answer 1

To find the area of a square represented by the equation \(3x^2 - 5\), we need to interpret this expression. The area \(A\) of a square is given by the square of the length of one of its sides. If we let the length of a side be represented by \(s\), then the area is:

\[
A = s^2
\]

In your case, the equation \(s^2 = 3x^2 - 5\) represents the area of the square. Thus:

\[
A = 3x^2 - 5
\]

This expression \(3x^2 - 5\) is already simplified, but we can analyze it further.

1. This expression represents the area of the square only if \(3x^2 - 5\) is non-negative, meaning \(3x^2 - 5 \geq 0\).
2. To find the values of \(x\) for which the area is non-negative, solve:

\[
3x^2 - 5 \geq 0
\]
\[
3x^2 \geq 5
\]
\[
x^2 \geq \frac{5}{3}
\]
\[
|x| \geq \sqrt{\frac{5}{3}} \approx 1.291
\]

Therefore, the area \(3x^2 - 5\) is non-negative (and thus a valid area for a square) for \(x \leq -\sqrt{\frac{5}{3}}\) and \(x \geq \sqrt{\frac{5}{3}}\).

When the expression is used to calculate the area of the square, \(A = 3x^2 - 5\) is the area in its simplest form. Depending on the context, if you have specific values of \(x\), you can plug them in to find a numerical area.

In conclusion, the area of the square is given by:

\[
\text{Area} = 3x^2 - 5
\]

with the condition \(3x^2 \geq 5\).