Problem 24 - Algebra for College Students

Question

Solve each equation. $$\frac{2}{x}+\frac{5}{x+2}=1$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5 + \sqrt{41}}{2}\) and \(x = \frac{5 - \sqrt{41}}{2}\).

Step 1: Identify a Common Denominator

The equations present fractions with denominators of \(x\) and \(x+2\). To solve for \(x\), identify a common denominator for these fractions. The common denominator is \(x(x+2)\).

Step 2: Rewrite Each Fraction

Rewrite each fraction to have the common denominator. We have: \(\frac{2}{x} = \frac{2(x+2)}{x(x+2)}\) and \(\frac{5}{x+2} = \frac{5x}{x(x+2)}\).

Step 3: Combine the Fractions

Combine the fractions: \(\frac{2(x+2) + 5x}{x(x+2)} = 1\). This results in a single fraction where the numerator is \(2(x+2) + 5x\).

Step 4: Clear the Denominator

Multiply both sides by the common denominator \(x(x+2)\) to clear the denominator: \(2(x+2) + 5x = x(x+2)\).

Step 5: Expand and Simplify

Expand and simplify both sides:- Left side: \(2x + 4 + 5x = 7x + 4\)- Right side: \(x^2 + 2x\).Equation: \(7x + 4 = x^2 + 2x\).

Step 6: Rearrange into a Quadratic Equation

Move all terms to one side to get a quadratic equation: \(x^2 + 2x - 7x - 4 = 0\) simplifies to \(x^2 - 5x - 4 = 0\).

Step 7: Solve the Quadratic Equation

We use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -5\), and \(c = -4\):- \(x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1}\)- Simplifying gives: \(x = \frac{5 \pm \sqrt{25 + 16}}{2}\)- Further simplifying: \(x = \frac{5 \pm \sqrt{41}}{2}\).

Step 8: Calculate the Solutions

The calculated solutions are \(x = \frac{5 + \sqrt{41}}{2}\) and \(x = \frac{5 - \sqrt{41}}{2}\).

Key Concepts

Fraction simplification

Fraction simplification involves reducing fractions to their simplest form. In this problem, the fractions \(\frac{2}{x}\) and \(\frac{5}{x+2}\) need to be analyzed and rewritten.
Fractions in mathematics often have variables as denominators, which can complicate their manipulation. To
simplify, you align the denominators through a series of steps that include rewriting each fraction with the common denominator and then combining them.By having \(\frac{2(x+2)}{x(x+2)}\) and \(\frac{5x}{x(x+2)}\), each fraction is expressed in terms of a common denominator, aiding in simplifying further calculations.
The goal is to remove complex variables from fractions by restructuring the equation to have a simplified
format.

Common denominators

Finding a common denominator is crucial for solving equations involving fractions. In this exercise, the
fractions \(\frac{2}{x}\) and \(\frac{5}{x+2}\) needed unifying under a common denominator for straightforward simplification.
The least common denominator of \(x\) and \(x+2\) is \(x(x+2) \).Here is why this is necessary:

• It allows fractions to be combined easily, which is needed when adding or subtracting fractions.
• It ensures consistency throughout the mathematical operations applied on both sides of the equation.

Using the common denominator \(x(x+2)\), each fraction can be rewritten, guiding the procedure to a more involved equation such as the single fraction own the left as \(\frac{2(x+2) + 5x}{x(x+2)}\). This is an essential step in clearing the path for further simplification.
The ultimate target is to simplify these fractions so you can then manipulate the rest of the equation more efficiently.

Quadratic formula

The quadratic formula is \[\x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]It is used when solving quadratic equations, such as \( x^2 - 5x - 4 = 0 \) derived from this problem. This formula is powerful because:

• It provides the exact solutions to any quadratic equation where real numbers exist.
• It utilizes the coefficients from the quadratic, identifying them as \( a = 1\), \( b = -5\), and \( c = -4\) in this specific problem.

By substituting these values into the quadratic formula, you can efficiently find the roots of the equation. The solutions are calculated as \(x = \frac{5 \pm \sqrt{41}}{2}\\). This comprehensive approach ensures that no potential solutions are overlooked.
Mastering this formula is essential for tackling quadratic equations across a variety of mathematical problems.

Rational equations

Rational equations are equations that involve fractions. The challenge with rational equations, like the one in
this exercise, is managing those fractions properly through solution steps. When handling rational equations:

• Finding a common denominator helps in eliminating fractions by allowing you to multiply through and clear denominators easily.
• Simplifying these equations is more straightforward when the fractions are rewritten in terms of a common denominator.
• Apply algebraic techniques like clearing denominators, cross-multiplication, and simplifying until you can isolate terms effectively.

In our exercise, solving \(\frac{2}{x} + \frac{5}{x+2} = 1\) required clear understanding and manipulation of the fractions by transforming it into a simplified quadratic form.
This paves the way for the quadratic formula to find the solutions. Rational equations necessitate meticulous care in simplification, ensuring that equivalencies hold true across transformations.