Problem 30 - Algebra 2

Question

Solve each equation for the given variable. $$ \frac{1}{c}-\frac{c}{a^{2}-b^{2}}=0 ; c $$

Step-by-Step Solution

Answer

The solutions for 'c' from the given equation are \( c=\sqrt{a^{2}-b^{2}} \) and \( c=-\sqrt{a^{2}-b^{2}} \).

Step 1: Understand the equation

The problem is to solve for variable 'c' from the equation \( \frac{1}{c}-\frac{c}{a^{2}-b^{2}}=0\). To do that, the first step is to try to eliminate fractions and simplify the equation.

Step 2: Clear fractions

The common denominator of the two fractions is \( c(a^{2} - b^{2}) \). Multiply each term by this common denominator to clear fractions: \( c(a^{2}-b^{2})\cdot\frac{1}{c}-(a^{2}-b^{2})\cdot\frac{c}{a^{2}-b^{2}}=0 \). Now simplify this equation to obtain: \( a^{2} - b^{2} - c^{2} = 0 \).

Step 3: Solve for 'c'

Rearrange the equation to solve for 'c': \( c^{2} = a^{2} - b^{2}\) . To get the value of 'c', take the square root on both sides of the equation. This gives two possible solutions: \( c = \sqrt{a^{2}-b^{2}} \) (positive root) and \( c = -\sqrt{a^{2}-b^{2}} \) (negative root)

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a vital tool for solving equations, especially when dealing with complex expressions. At its core, it involves rearranging and simplifying equations to isolate the variable you want to solve.

In the given exercise, the goal was to isolate 'c'. To achieve this, the equation had to be reshaped from an initial format with fractions to a cleaner, quadratic format.

• Identifying equivalent expressions and changing their form can make it easier to see solutions.
• Using operations like addition, subtraction, multiplication, and division helps in transforming parts of the equation.
• Pay attention to symmetry and structure; this can sometimes suggest substitutions or strategies for simplification.

Effective algebraic manipulation lays the foundation for finding solutions by simplifying an equation to its most basic components.

Fractions in Equations

Fractions can complicate equations, but understanding how to manage them is crucial for solving problems. The key challenge in dealing with fractions is often to eliminate them to make the equation easier to work with.

The exercise involves clearing the fractions in the equation \( \frac{1}{c} - \frac{c}{a^{2} - b^{2}} = 0 \).

• The first step is to find a common denominator, which helps in combining terms. For this equation, the common denominator is \( c(a^{2} - b^{2}) \).
• Multiplying each term by this denominator eliminates the fractions, reducing the equation to a simpler form: \( a^{2} - b^{2} - c^{2} = 0 \).

This method allows you to shift from a fraction-based equation to one that is more straightforward to solve, showing how powerful removing fractions can be in algebraic equations.

Quadratic Equations

Quadratic equations are equations of the form \( ax^{2} + bx + c = 0 \). They are fundamental in algebra and come up frequently. The goal is to find the values of \( x \) that satisfy the equation.

In this exercise, the simplified equation \( a^{2} - b^{2} - c^{2} = 0 \) is a quadratic form in terms of \( c \).

• This type of equation can have up to two solutions because of its squared variable.
• Here, solving the quadratic \( c^{2} = a^{2} - b^{2} \) involves taking the square root of both sides, leading to two potential solutions: \( c = \sqrt{a^{2} - b^{2}} \) and \( c = -\sqrt{a^{2} - b^{2}} \).
• Always consider both the positive and negative roots when dealing with square roots in quadratic equations.

Using these methods reliably yields the potential solutions \( c \) must satisfy, showcasing the elegance of algebra in solving quadratic problems.