Problem 14 - Algebra 2

Question

Solve each equation. Check each solution. $$ \frac{1}{x}+\frac{x}{2}=\frac{x+4}{2 x} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1\).

Step 1: Multiply each term by \(2x\)

Multiply each term by \(2x\) to eliminate the denominator. The new equation will be: \(2 + x^2 = x + 4\)

Step 2: Rearrange the Equation

Subtract \(x\) from each side to isolate the \(x^2\) term: \(x^2 - x + 2 = 4\). Then subtract 2 from both sides to find: \(x^2 - x = 2\). Now the equation can be rearranged to the form \(x^2 - x - 2 = 0\)

Step 3: Factor the Equation

Factor the equation to solve for \(x\). (\(x-2)(x+1) = 0\). Set each factor equal to zero and solve for \(x\): \(x - 2 = 0 \Longrightarrow x = 2\) and \(x + 1 = 0 \Longrightarrow x = -1\)

Step 4: Check each solution

Substitute each solution back into the original equation and check. For \(x = 2\): \(\frac{1}{2} + 1 = 2 \Longrightarrow 0.5 + 1 = 1.5 \neq 2\) thus \(x = 2\) is not a solution. For \(x = -1\): \(\frac{-1}{1} + \frac{-1}{2} = \frac{-1 -4}{-2} \Longrightarrow -1 - 0.5 = 2.5 = 2.5\) thus \(x = -1\) is a solution.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that are fundamental in algebra. They are typically written in the standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The key feature of a quadratic equation is the \( x^2 \) term, which creates a parabolic graph when plotted.
Quadratic equations can be identified by the highest exponent of the variable being 2.

• They often have two solutions, although sometimes these solutions can be complex or imaginary numbers.
• Simplifying to this form helps in identifying the methods suitable for solving.

Understanding quadratic equations is crucial as they frequently appear in various areas of math and real-world applications.

Solving Equations

When solving equations, the main goal is to find the values of the variables that make the equation true.
For equations involving fractions, like rational equations, it is essential to clear the fractions first for simplicity.
In the exercise, multiplying each term by the least common denominator \( 2x \) helps in eliminating the fractions, resulting in a simpler equation: \( 2 + x^2 = x + 4 \).

• Eliminating fractions is critical for simplifying the problem into a manageable quadratic equation.
• Rearranging terms to one side is the next step, resulting in a standard form quadratic equation, allowing factoring or other methods.

By systematically simplifying and reorganizing the terms, solving becomes more straightforward.

Checking Solutions

Checking solutions is an essential final step in solving equations, especially for verifying the validity of potential solutions.
This process ensures that the solutions satisfy the original equation.

• For \( x = 2 \), plugging back into the original equation results in a false statement, indicating it was an extraneous solution.
• For \( x = -1 \), substituting back into the original equation confirms it satisfies the equation.

Extraneous solutions can occur when multiplying through by terms that might be zero, thus verifying the solutions prevents errors in final answers.

Factoring Polynomials

Factoring polynomials is a technique used to break down a polynomial into simpler components, making it easier to solve.
The goal is to express the polynomial as a product of its factors.
For the quadratic \( x^2 - x - 2 \), factoring results in \((x-2)(x+1) = 0\).

• This factorization step is crucial for finding the roots of the equation.
• Setting each factor equal to zero provides the solutions \( x=2 \) and \( x=-1 \).

Factoring is a primary method because it directly reveals solutions for quadratic and higher-degree equations.