What is the solution of the equation $x^2=64$?
The given equation is \(x^2 = 64\). This is a quadratic equation where we need to find the real number(s) \(x\) that, when multiplied by themselves, result in 64. Since 64 is a positive number, there will be two solutions to this equation.
To solve for \(x\), we take the square root of both sides of the equation:
\[\sqrt{x^2} = \sqrt{64}\]
Recall that for any real number \(x\), the expression \(\sqrt{x^2}\) is equal to the absolute value of \(x\), written as \(|x|\). Therefore, the equation becomes:
\[|x| = 8\]
This means that the distance of \(x\) from zero on the number line is 8. Thus, \(x\) can be 8 or -8. We can write this compactly as:
\[x = \pm 8\]
Comparing our result \(x = \pm 8\) with the given choices:
The correct option is d.
In everyday life, square roots are used in many geometric calculations. For example, if you know the area of a square garden is 64 square meters, you can find the length of one side of the garden by taking the square root of the area: \(\sqrt{64} = 8\) meters. However, when working with algebraic equations like \(x^2 = 64\), we must consider both positive and negative roots because \((-8)^2\) also equals 64. Understanding this ensures you don't miss solutions when modeling physical or abstract problems.