Problem 53 - Basic Technical Mathematics with Calculus

Question

Solve the given equations involving fractions. $$\frac{1}{2 x}-\frac{3}{4}=\frac{1}{2 x+3}$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -2\).

Step 1: Eliminate Fractions

To solve the equation \( \frac{1}{2x} - \frac{3}{4} = \frac{1}{2x+3} \), first eliminate the fractions by finding a common denominator. The denominators are \(2x\), \(4\), and \(2x+3\). The common denominator is \(4(2x)(2x+3)\). Multiply every term by this common denominator to clear the fractions.

Step 2: Simplify the Equation

After multiplying every term by the common denominator, the equation simplifies to \(4(2x+3) - 3(2x)(2x+3) = 4(2x)\). Expand each term: \(8x + 12 - 12x^2 - 18x = 8x\).

Step 3: Rearrange Terms

Collect like terms on one side of the equation: \(-12x^2 - 18x + 8x + 12 = 8x\). Simplify to \(-12x^2 - 10x + 12 = 8x\).

Step 4: Solve for x

Move all terms to one side of the equation, \(-12x^2 - 10x + 12 - 8x = 0\), simplifying to \(-12x^2 - 18x + 12 = 0\). Factor the quadratic equation: \(-6(2x^2 + 3x - 2) = 0\). Further factor to \(-6(2x - 1)(x + 2) = 0\).

Step 5: Find Solutions for x

The solutions to \(-6(2x - 1)(x + 2) = 0\) are \(2x - 1 = 0\) (i.e., \(x = \frac{1}{2}\)) and \(x + 2 = 0\) (i.e., \(x = -2\)). Check each solution in the original equation.

Key Concepts

Fractions

Fractions represent parts of a whole, often using a numerator (top number) and a denominator (bottom number). In this exercise, we dealt with several fractions, such as \( \frac{1}{2x} \), \( \frac{3}{4} \), and \( \frac{1}{2x+3} \). These fractions make equations more complex, so eliminating them is often the first step in solving such equations.

Working with fractions typically involves steps like:

• Identifying the least common denominator, which helps align the fractions for addition, subtraction, or elimination.
• Converting fractions to have the common denominator if needed.
• Adding, subtracting, or otherwise manipulating these fractions to simplify the equation.

Understanding fractions and how to eliminate them is essential in reducing the complexity of equations when solving. It often requires multiplying each term of the equation by a common multiple of the denominators to effectively "clear" the fractions.

Common Denominator

The concept of a common denominator is a core component when dealing with equations involving fractions. It allows us to write the fractions with equivalent terms for easy arithmetic manipulation. In our example, the equation \( \frac{1}{2x} - \frac{3}{4} = \frac{1}{2x+3} \) included different denominators: \(2x\), \(4\), and \(2x+3\).

Finding the common denominator involves:

• Identifying all unique denominators in the equation.
• Determining a common multiple of all these denominators, often by multiplying them together.

The aim is to rewrite each term of the equation with this common denominator, which simplifies elimination of fractions and enables comparative or combinative equality. Once terms share a common denominator, it's straightforward to combine or eliminate them, simplifying the equation significantly.

Quadratic Equation

A quadratic equation is any equation that can be rearranged to the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our solution, after clearing the fractions and simplifying, we ended with a quadratic equation \(-12x^2 - 18x + 12 = 0\).

This form is crucial because:

• It indicates the equation will have two solutions or roots.
• It's a well-known form making it easier to apply various solving techniques like factoring or using the quadratic formula.

Quadratics frequently appear in algebra due to their prominence in modeling curves and other significant mathematical behaviors.

Factoring

Factoring is a mathematical process used to simplify expressions or solve equations. Specifically, it involves expressing a polynomial as a product of simpler polynomials or numbers. In our solution, we eventually factored \(-12x^2 - 18x + 12 = 0\) to \(-6(2x-1)(x+2) = 0\).

The factoring process includes:

• Looking for common factors in all terms.
• Using patterns such as the difference of squares or trinomial squares.
• Breaking the quadratic trinomial into a product of two binomials, as was done here.

Factoring transforms complex expressions into simpler multiplicative forms, making it straightforward to solve for unknown variables. Solving the factors involves setting each term to zero, which provides the roots of the equation, thus completing the solution process.