Problem 120 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$x=s^{2} \quad \text { for } s$$

Step-by-Step Solution

Answer

\(s = \pm\sqrt{x}\)

Step 1: Identify the Formula

We are given the formula \(x = s^2\). Our task is to solve for \(s\). This means we need to express \(s\) in terms of \(x\).

Step 2: Isolate the Variable

To isolate \(s\), take the square root of both sides of the equation. This will give us \(s = \pm\sqrt{x}\) since taking the square root of both sides introduces both the positive and negative square roots.

Key Concepts

Isolating Variables

Isolating variables is a fundamental step in solving equations. It involves rearranging the equation to make one variable the subject, which means having the variable by itself on one side of the equation. This allows us to see that variable in terms of the other elements present in the equation. In the given exercise, we started with the formula:

• \(x = s^2\)

Our goal here was to isolate the variable \(s\). To achieve this:

• We knew that \(s\) must be alone on one side of the equation.
• This required identifying the operation connecting \(s\) to \(x\), in this case, squaring.

After identifying that \(s\) was squared, the next logical step was to cancel out or "reverse" this operation, which naturally leads us to the square root function.

Square Roots

The square root is one of the inverse operations of squaring a number. When you square a number, you multiply it by itself. To revert this process, you take the square root. In mathematics, if \(a^2 = b\), then \(a = \pm\sqrt{b}\). The plus or minus sign (\(\pm\)) indicates that there can be two potential solutions for \(a\), because both positive and negative numbers, when squared, yield a positive result.

In our case:

• To isolate \(s\) in \(x = s^2\), we need to determine \(s\) by taking the square root of both sides.
• This gave us: \(s = \pm\sqrt{x}\).

Remember, taking the square root of a squared variable brings back both possible values: positive and negative, since \((s^2 = x)\) can stem from both \(s = \sqrt{x}\) and \(s = -\sqrt{x}\).

When dealing with square roots, always remember they imply both directions unless specified otherwise.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations and expressions to simplify and solve for variables or to find equivalent expressions. In algebra, several techniques can be applied such as:

• Adding, subtracting, multiplying, or dividing both sides of an equation by the same number.
• Applying functions, like square roots or exponentiation, to both sides.

In the given problem, we specifically applied algebraic manipulation by using the inverse operation of squaring, which was taking the square root.

• Starting from \(x = s^2\), to solve for \(s\), we performed the inverse operation of squaring.
• This transforming action allowed us to obtain the expression \(s = \pm\sqrt{x}\), repositioning the variable \(s\) isolated on one side.

Algebraic manipulation is essential to solving and simplifying expressions, giving clarity to complex mathematical problems. By understanding and executing these manipulations correctly, one can master the process of finding solutions effectively.