Consider This Rational Equation:${ \frac{1}{x} + \frac{1}{x-2} = \frac{1}{4} }$Use The Least Common Denominator To Simplify The Rational Equation Into A Standard Form Quadratic Equation. Replace The Values Of { B$}$ And

Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of the underlying mathematics. In this article, we will explore the process of solving rational equations using the least common denominator (LCD) method. We will use the given rational equation $\frac{1}{x} + \frac{1}{x-2} = \frac{1}{4}$ as an example to demonstrate the steps involved in simplifying the equation into a standard form quadratic equation.

Understanding Rational Equations

A rational equation is an equation that contains one or more rational expressions, which are expressions that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are polynomials and $q$ is not equal to zero. Rational equations can be linear or quadratic, and they can be solved using various methods, including the LCD method.

The Least Common Denominator (LCD) Method

The LCD method is a technique used to simplify rational equations by finding the least common multiple of the denominators of the rational expressions involved. The LCD is the smallest expression that is divisible by all the denominators of the rational expressions. Once the LCD is found, the rational expressions can be rewritten with the LCD as the denominator, and the equation can be simplified.

Simplifying the Rational Equation

To simplify the given rational equation $\frac{1}{x} + \frac{1}{x-2} = \frac{1}{4}$, we need to find the LCD of the denominators $x$ and $x-2$. The LCD is $4(x)(x-2)$, which is the smallest expression that is divisible by both $x$ and $x-2$.

Rewriting the Rational Expressions

Now that we have found the LCD, we can rewrite the rational expressions with the LCD as the denominator. The first rational expression $\frac{1}{x}$ can be rewritten as $\frac{4(x-2)}{x(x-2)}$, and the second rational expression $\frac{1}{x-2}$ can be rewritten as $\frac{4x}{x(x-2)}$. The equation becomes:

$$\frac{4(x-2)}{x(x-2)} + \frac{4x}{x(x-2)} = \frac{1}{4}$$

Simplifying the Equation

Now that the rational expressions have been rewritten with the LCD as the denominator, we can simplify the equation by combining the fractions. Since the denominators are the same, we can add the numerators:

$$\frac{4(x-2) + 4x}{x(x-2)} = \frac{1}{4}$$

Multiplying Both Sides by the LCD

To eliminate the fractions, we can multiply both sides of the equation by the LCD, which is $x(x-2)$. This will give us a quadratic equation in standard form:

$$4(x-2) + 4x = \frac{x(x-2)}{4}$$

Expanding and Simplifying

Now that we have multiplied both sides of the equation by the LCD, we can expand and simplify the equation:

$$4x - 8 + 4x = \frac{x^2 - 2x}{4}$$

Multiplying Both Sides by 4

To eliminate the fractions, we can multiply both sides of the equation by 4:

$$16x - 32 = x^2 - 2x$$

Rearranging the Equation

Now that we have multiplied both sides of the equation by 4, we can rearrange the equation to put it in standard form:

$$x^2 - 18x + 32 = 0$$

Solving the Quadratic Equation

The quadratic equation $x^2 - 18x + 32 = 0$ can be solved using various methods, including factoring, the quadratic formula, or graphing. In this case, we can use the quadratic formula to find the solutions:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Finding the Solutions

Now that we have the quadratic formula, we can plug in the values of $a$, $b$, and $c$ to find the solutions:

$$x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(1)(32)}}{2(1)}$$

Simplifying the Solutions

Now that we have plugged in the values, we can simplify the solutions:

$$x = \frac{18 \pm \sqrt{324 - 128}}{2}$$

Simplifying the Solutions (continued)

$$x = \frac{18 \pm \sqrt{196}}{2}$$

Simplifying the Solutions (continued)

$$x = \frac{18 \pm 14}{2}$$

Finding the Solutions (continued)

Now that we have simplified the solutions, we can find the two possible values of $x$:

$$x = \frac{18 + 14}{2} \quad \text{or} \quad x = \frac{18 - 14}{2}$$

Finding the Solutions (continued)

$$x = \frac{32}{2} \quad \text{or} \quad x = \frac{4}{2}$$

Finding the Solutions (continued)

$$x = 16 \quad \text{or} \quad x = 2$$

Conclusion

In this article, we have demonstrated the process of solving a rational equation using the least common denominator (LCD) method. We have used the given rational equation $\frac{1}{x} + \frac{1}{x-2} = \frac{1}{4}$ as an example to show the steps involved in simplifying the equation into a standard form quadratic equation. We have found the solutions to the quadratic equation using the quadratic formula and have simplified the solutions to find the two possible values of $x$.