Problem 39 - Elementary Algebra

Question

For the following problems, solve for the indicated variable. $$ x^{2}=9 b^{2}, \text { for } x $$

Step-by-Step Solution

Answer

Question: Solve for x in the equation \(x^2 = 9b^2\). Answer: x = ±3b.

Step 1: Isolate x squared term

Equation: \(x^2 = 9b^2\). x squared term is already isolated.

Step 2: Simplify to find x

To solve for x, we need to take the square root of both sides of the equation. $$ \sqrt{x^2} = \sqrt{9b^2} $$ This gives us: $$ x = \pm\sqrt{9b^2} $$

Step 3: Factor and simplify the expression

Let's factor 9 from the expression inside the square root. $$ x = \pm\sqrt{9}*\sqrt{b^2} $$ Now we can further simplify by taking the square root of 9: $$ x = \pm3*\sqrt{b^2} $$

Step 4: Final Answer

Consequently, x is equal to: $$ x = \pm3b $$

Key Concepts

Solving Equations

Solving equations is like solving a puzzle. When you see an equation such as \(x^2 = 9b^2\), it means you have to find the value of \(x\) that makes the equation true. Here is what you need to keep in mind:

• Identify where your unknown (or variable), like \(x\), is in the equation.
• Your goal is to "solve for \(x\)," which means making \(x\) the subject of the equation.
• To solve it, apply mathematical operations to "undo" whatever is being done to \(x\).

For this specific problem, since \(x^2 = 9b^2\), we are looking at a squared variable. Solving this involves reversing the squaring operation, which is commonly done by taking the square root.

Square Roots

Square roots are the opposite of squaring a number. When we take the square root of a number, we are looking for a value that, when multiplied by itself, gives us the original number.

For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\). In the equation \(x^2 = 9b^2\), we take the square root of both sides:

• Square root of \(x^2\) is \(x\), because squaring and square rooting cancel each other out.
• Square root of \(9b^2\) is \(\pm3b\), which includes both positive and negative roots.

Consider the positive and negative solutions because squaring either a positive or a negative number results in a positive value. This is why when solving \(x^2 = a\), we write \(x = \pm\sqrt{a}\).

Variable Isolation

Variable isolation is a key strategy. The idea is to rearrange the equation to get the variable \(x\) by itself on one side of the equation. Let's see how it's applied:

• Begin with the equation \(x^2 = 9b^2\), the \(x^2\) term is already isolated, meaning it’s already by itself on one side of the equation.
• The next step is simplifying this equation, which involves taking the square root of both sides.
• After taking the square root, you get \(x = \pm3b\), which shows that \(x\) is now isolated.

By isolating the variable, you simplify the problem down to an answer that shows the variable directly, making it clear what \(x\) can be. Remember, the goal is to untangle the variable from everything else.