Problem 39 - Precalculus Mathematics for Calculus

Question

Solve the equation for the indicated variable. $$a^{2}+b^{2}=c^{2} ; \quad \text { for } b$$

Step-by-Step Solution

Answer

\( b = \pm \\sqrt{c^2 - a^2} \)

Step 1: Isolate the b² Term

To begin solving the equation for \( b \), we need to isolate the \( b^2 \) term on one side of the equation. The original equation is \( a^2 + b^2 = c^2 \). Subtract \( a^2 \) from both sides to get: \[ b^2 = c^2 - a^2 \]

Step 2: Solve for b

Now that we have \( b^2 \) isolated, the next step is to solve for \( b \). Take the square root of both sides of the equation to solve for \( b \). Be sure to consider both the positive and negative square roots: \[ b = \sqrt{c^2 - a^2} \] or \[ b = -\sqrt{c^2 - a^2} \] These represent the positive and negative solutions for \( b \).

Key Concepts

Isolation of Variables

Isolating a variable is a crucial step in solving equations. It involves manipulating the equation to express one particular variable alone on one side. This process allows us to solve for the variable of interest easily.

When isolating a variable, like in our exercise where we isolate \( b^2 \), the goal is to perform algebraic operations that unravel the equation. These operations can include:

• Addition or subtraction to move terms from one side to the other
• Multiplication or division to simplify coefficients
• Using inverse operations to simplify the equation

In the original equation \( a^2 + b^2 = c^2 \), we subtract \( a^2 \) from both sides to achieve \( b^2 = c^2 - a^2 \). This manipulation places \( b^2 \) by itself, isolated, making the equation ready for further solving steps. Remember, the key is to reverse any operations affecting the variable until it stands alone on one side of the equation.

Quadratic Equations

Quadratic equations are central in algebra and often appear as equations of the form \( ax^2 + bx + c = 0 \). They are called quadratic because they involve terms that are raised to the power of two. However, in our exercise, the equation takes a slightly different form without a linear term, focusing instead on \( b^2 \).

The hallmark of quadratic equations is their parabolic graphs when plotted and the presence of up to two solutions. These solutions arise because taking the square root of both sides in a quadratic equation yields both positive and negative results.

Quadratic equations can be tackled with several methods:

• Factoring: Breaking down the equation into a product of simpler expressions if possible
• Completing the square: Rewriting the equation in a perfect square form
• Quadratic formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for general solutions

Understanding the nature of quadratic equations helps in solving them more effectively, as seen in the step-by-step approach to our specific exercise.

Square Root Property

The square root property is a mathematical tool useful whenever we encounter expressions or equations involving squares. It states that if \( x^2 = k \), then \( x = \sqrt{k} \) or \( x = -\sqrt{k} \), indicating both possible solutions.

In the exercise, once we isolate \( b^2 \) as \( b^2 = c^2 - a^2 \), we employ the square root property to solve for \( b \). Applying this property means taking the square root of the entire expression on the right-hand side.

Important points about the square root property include:

• Both the positive and negative roots must be considered, which can yield two solutions
• The property is a direct route to solving quadratic expressions of the form \( b^2 = k \)
• It simplifies the process of solving equations by reducing them to linear forms

Employing the square root property is essential in many algebraic problems and offers a straightforward method to solve for squared terms, as demonstrated in deriving the values for \( b \) in this situation.