Step 1: Identify the equation.
We have an equation in the form \( x^2 = a \), where \( a \) is a positive real number.
Step 2: Take the square root of both sides.
To isolate \( x \), we take the square root of both sides of the equation because the square root is the inverse operation of squaring. So we have:
\[
\sqrt{x^2} = \sqrt{160}
\]
This yields two possible solutions, one positive and one negative, because both a positive and a negative number squared will give a positive result.
Step 3: Solve for the positive solution.
\( x_1 = +\sqrt{160} \)
Step 4: Simplify the square root, if possible.
\( x_1 = +\sqrt{16 \times 10} \)
Since \( \sqrt{16} = 4 \),
\( x_1 = +4\sqrt{10} \)
Step 5: Solve for the negative solution.
\( x_2 = -\sqrt{160} \)
Following the same simplification:
\( x_2 = -4\sqrt{10} \)
Therefore, the two solutions to the equation \( x^2 = 160 \) are:
\( x_1 = +4\sqrt{10} \)
and
\( x_2 = -4\sqrt{10} \)