Problem 136 - College Algebra

Question

Exercises \(134-136\) will help you prepare for the material covered in the next section. $$\text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x}$$

Step-by-Step Solution

Answer

The solutions of the equation are \(x = 3\) and \(x = -1\).

Step 1: Clear the Denominator and Simplify

First, we'll multiply the entire equation by the denominators 4x + 3 and x to eliminate them. The equation becomes: x(x+2) = 4x + 3

Step 2: Expand and Rearrange

Expand the left side of the equation: \(x^2 + 2x = 4x + 3\). Then, rearrange the equation by setting it equal to 0: \(x^2 + 2x - 4x - 3 = 0\) simplifies to \(x^2 - 2x - 3 = 0\)

Step 3: Solve for x

This is a quadratic equation in the form \(ax^2 + bx + c = 0\). We can find the roots of the equation using the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)] / 2a\). Substituting a=1, b=-2 and c=-3 into the formula gives \(x = [2 ±sqrt((-2)^2 - 4*1*(-3))] / 2*1\) which simplifies to \(x = 1 ± 2\). This results in two solutions for x: x=3 and x=-1.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra, represented by the formula \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable you need to solve for. Quadratics have distinctive characteristics:

• The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).
• A standard quadratic equation has two solutions, which can be real or complex numbers.
• The solutions are found where the parabola intersects the \( x \)-axis, also known as the roots of the equation.

Understanding quadratic equations is crucial as they form the basis for many real-world applications, from physics to economics.

Clearing Denominators

When solving rational equations, one of the first steps is clearing the denominators to simplify the equation. This involves multiplying both sides of the equation by the least common denominator (LCD).

• In the equation, \( \frac{x+2}{4x+3}=\frac{1}{x} \), the LCD is \( (4x+3)x \).
• Multiply both sides of the equation by this LCD: \( x(x+2) = 4x + 3 \).
• This step removes the fractions from the equation, making it easier to solve.

Clearing denominators is a valuable technique for simplifying complex rational expressions, allowing you to focus on solving the core equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here’s how it works:

• \( b^2 - 4ac \) is known as the discriminant and determines the nature of the roots. A positive discriminant indicates real and distinct roots, while a zero discriminant indicates a double root.
• Plug in the values of \( a \), \( b \), and \( c \) from your quadratic equation to find the roots.
• For example, in the equation \( x^2 - 2x - 3 = 0 \), substitute \( a = 1 \), \( b = -2 \), \( c = -3 \) into the formula.
• This results in two solutions for \( x \): \( x = 3 \) and \( x = -1 \).

The quadratic formula is essential for finding exact solutions when factoring is complex or impossible.

Step-by-Step Problem Solving

Solving equations step-by-step ensures clarity and accuracy in your work. Here is a structured approach:

• Identify the type of equation (e.g., rational, linear, quadratic).
• Clear denominators, if present, to simplify the approach to solving.
• Rearrange the equation to standard form, if necessary, similar to \( ax^2 + bx + c = 0 \).
• Use appropriate methods such as factoring or the quadratic formula to find solutions.
• Always double-check the solutions by substituting them back into the original equation to verify correctness.

This methodical approach to problem-solving helps break down complex math problems into manageable parts, facilitating a deeper understanding and more reliable solutions.