Problem 93 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Isolate the Variable Term

We start with the equation \[ a^2 + b^2 = c^2 \]To solve for \( b \), we need to isolate \( b^2 \) by subtracting \( a^2 \) from both sides of the equation, as follows:\[ b^2 = c^2 - a^2 \]

Step 2: Solve for the Variable

To solve for \( b \), take the square root of both sides of the equation. This will give us two potential solutions because the square root function has a positive and negative output:\[ b = \pm \sqrt{c^2 - a^2} \]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry. It relates the lengths of the sides of a right triangle. Specifically, it states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms, if a triangle has sides of lengths \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse), the theorem is written as:\[ a^2 + b^2 = c^2 \] This theorem is useful because it allows you to find an unknown side of a right triangle if you know the lengths of the other two sides. Initially developed by the ancient Greeks, it has applications not just in geometry but also in numerous fields like physics, engineering, and computer science.

Isolating Variables

When solving equations, isolating the variable you are solving for is a crucial step. This means getting the variable by itself on one side of the equation. To do this, you perform operations to move other numbers or terms to the opposite side.In our example of the Pythagorean theorem \( a^2 + b^2 = c^2 \), if we're solving for \( b \), we need to isolate \( b^2 \). This is done by subtracting \( a^2 \) from both sides, leading to:\[ b^2 = c^2 - a^2 \] Remember these tips:

• Perform the same operation on both sides to maintain equality.
• Be conscious of the order of operations: parenthesis, exponents, multiplication/division, and addition/subtraction (PEMDAS).
• Check your solution by substituting back in to the original equation.

Isolating variables helps simplify problems and is a key skill in algebra and other branches of mathematics.

Square Roots

Square roots are numbers which produce a specific number when multiplied by themselves. The operation of taking a square root is essentially the inverse of squaring a number.Consider our equation from the Pythagorean theorem, after isolating \( b^2 \):\[ b^2 = c^2 - a^2 \] To solve for \( b \), we must take the square root of \( b^2 \). Because the square of a number has two possible roots (both positive and negative), the solution includes both signs:\[ b = \pm \sqrt{c^2 - a^2} \] A few points on square roots:

• The symbol \( \sqrt{} \) denotes the principal (positive) square root.
• Negative roots are also solutions, hence the reason for "\( \pm \)" in equations.
• Always check if your square root is simplified to its simplest form.
• In real-world applications, the context will often determine if a negative root is meaningful.

Understanding square roots is vital as it appears in many areas of mathematics, physics and even in financial calculations.