Problem 25 - Algebra 1

Question

Solve the equation by multiplying each side by the least common denominator. $$2+\frac{8}{x-5}=\frac{x+5}{x^{2}-25}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\) and \(x = 7\). They are the only solutions that do not make the original equation undefined.

Step 1: Identify the least common denominator

First, identify the least common denominator (LCD). In this case, that is \(x^{2}-25\). This is because it's the only denominator that isn't already factored into the others.

Step 2: Multiply each side by the least common denominator

Multiply each side by \(x^{2}-25\) to eliminate the fractions: \((2+(8/(x-5))) * (x^{2}-25)=(x+5)\). Simplify this to \(2x^{2}-50 + 8x - 40 = x^{3}+5x^{2}-25x-125\).

Step 3: Simplify the equation and solve for x

Rearrange and simplify the equation to set it equal to zero: \(x^{3}+5x^{2}-25x-125 -2x^{2}+50 - 8x +40=0\). This simplifies to \(x^{3} + 3x^{2} - 33x - 35 = 0\). By solving this equation, you get three possible solutions for x: \(x = -5, x = 1, x = 7\). However, the solution \(x = -5\) should be discarded because it would make the original equation undefined.

Key Concepts

Least Common Denominator

Understanding the least common denominator (LCD) is crucial when solving rational equations. It helps to eliminate fractions, making equations simpler to solve.
The LCD is the smallest expression that all the denominators in the equation can divide into evenly.
To find the LCD, start by factoring each denominator. Often, examining the structure of each term helps find a common pattern.

• For the exercise, the denominators are \(x-5\) and \(x^2 - 25\). Notice that \(x^2 - 25\) can be factored into \((x-5)(x+5)\).
• Since \(x^2 - 25\) already contains \(x-5\), the LCD is \(x^2 - 25\).

Using the LCD in the equation lets us clear out the fractions by multiplying every term by this common denominator.

Simplifying Equations

Simplifying an equation involves removing complex terms and reducing it to its simplest form. This process is essential for solving rational equations, as it makes them more manageable.
Once you've multiplied each side of the equation by the LCD, you eliminate the fractions. This results in a polynomial equation, which is easier to work with.

• Start by expanding the equation. For example, using our multiplied equation by the LCD, \((2 + \frac{8}{x-5})(x^2-25) = (x+5)\).
• Expand and simplify: \(2x^2 - 50 + 8x - 40 = x^3 + 5x^2 - 25x - 125\).

Combine like terms to further reduce the equation so you can isolate the variable \(x\) to find possible solutions.

Undefined Expressions

When solving rational equations, it's important to check for solutions that make any part of the equation undefined.
An expression is undefined when a denominator equals zero, as division by zero is not possible. We must exclude any solutions that result in zero denominators.

• For the given exercise, look at the denominators \(x-5\) and \(x^2-25\).
• The expressions are undefined for \(x = 5\) and \(-5\), since these values result in a zero denominator, leading to division by zero.

Always test your solutions against the original equation to ensure they do not result in undefined expressions, keeping the solutions that remain valid.