Solve for $x$ in the equation $x^2=121$
The equation to solve is \(x^2 = 121\). We are looking for all real numbers \(x\) such that their square is equal to 121. Recall that the square of a number \(x\), denoted by \(x^2\), is the result of multiplying \(x\) by itself. If \(x^2 = c\) for some positive constant \(c\), then \(x\) can be either the positive square root of \(c\) or the negative square root of \(c\). In this case, \(c = 121\).
We can rewrite the equation as \(x^2 - 121 = 0\). This expression is a difference of squares, which follows the pattern \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x\) and \(b = \sqrt{121} = 11\). Thus, we have \(x^2 - 11^2 = (x - 11)(x + 11) = 0\).
To satisfy the equation \((x - 11)(x + 11) = 0\), one of the factors must be zero. This gives us two possible cases:
Therefore, the solutions to the equation are \(x = 11\) and \(x = -11\). Comparing these results to the provided options, the correct choice is \(x = 11, -11\).
Solving quadratic equations like \(x^2 = 121\) is essential in physics and engineering. For example, if you are calculating the area of a square-shaped garden with an area of 121 square meters, the length of the side must be \(x = \sqrt{121} = 11\) meters (since length cannot be negative). Understanding how to handle both positive and negative roots is fundamental for solving more complex geometric and physical problems where variables represent physical dimensions or coordinate positions.