Solve The Rational Equation By Multiplying Both Sides By The LCD. Check Your Results For Extraneous Solutions.${ \frac{3}{x^2+5x+6} + \frac{x-1}{x+2} = \frac{7}{x+3} }$1. { X = \square $}$ Is A Solution.2. [$ X = \square

Introduction

Rational equations are a type of algebraic equation that involves fractions with polynomials in the numerator and denominator. Solving rational equations can be challenging, but one common method is to multiply both sides of the equation by the least common denominator (LCD). In this article, we will explore how to solve a rational equation by multiplying both sides by the LCD and check for extraneous solutions.

Understanding Rational Equations

A rational equation is an equation that contains one or more fractions with polynomials in the numerator and denominator. Rational equations can be written in the form:

${ \frac{p(x)}{q(x)} + \frac{r(x)}{s(x)} = \frac{t(x)}{u(x)} }$

where p(x), q(x), r(x), s(x), t(x), and u(x) are polynomials.

Finding the Least Common Denominator (LCD)

To solve a rational equation, we need to find the least common denominator (LCD) of the fractions. The LCD is the smallest polynomial that is a multiple of all the denominators in the equation. To find the LCD, we need to factor each denominator and identify the common factors.

Example: Finding the LCD

Consider the rational equation:

${ \frac{3}{x^2+5x+6} + \frac{x-1}{x+2} = \frac{7}{x+3} }$

To find the LCD, we need to factor each denominator:

${ x^2+5x+6 = (x+2)(x+3) }$ ${ x+2 = x+2 }$ ${ x+3 = x+3 }$

The LCD is the product of the common factors:

${ LCD = (x+2)(x+3) }$

Multiplying Both Sides by the LCD

Once we have found the LCD, we can multiply both sides of the equation by the LCD to eliminate the fractions. This will give us a polynomial equation that we can solve.

Example: Multiplying Both Sides by the LCD

Consider the rational equation:

${ \frac{3}{x^2+5x+6} + \frac{x-1}{x+2} = \frac{7}{x+3} }$

We have already found the LCD:

${ LCD = (x+2)(x+3) }$

We can multiply both sides of the equation by the LCD:

${ (x+2)(x+3) \left( \frac{3}{x^2+5x+6} + \frac{x-1}{x+2} \right) = (x+2)(x+3) \left( \frac{7}{x+3} \right) }$

This simplifies to:

${ 3(x+3) + (x-1)(x+2) = 7(x+2) }$

Simplifying the Equation

We can simplify the equation by combining like terms:

${ 3x+9 + x^2+x-2 = 7x+14 }$

This simplifies to:

${ x^2+4x+7 = 7x+14 }$

Solving the Equation

We can solve the equation by moving all the terms to one side:

${ x^2-3x-7 = 0 }$

This is a quadratic equation, and we can solve it using the quadratic formula:

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

where a, b, and c are the coefficients of the quadratic equation.

Example: Solving the Quadratic Equation

Consider the quadratic equation:

${ x^2-3x-7 = 0 }$

We can plug in the values of a, b, and c into the quadratic formula:

${ x = \frac{-(-3) \pm \sqrt{(-3)^2-4(1)(-7)}}{2(1)} }$

This simplifies to:

${ x = \frac{3 \pm \sqrt{9+28}}{2} }$

This simplifies to:

${ x = \frac{3 \pm \sqrt{37}}{2} }$

Checking for Extraneous Solutions

When we multiplied both sides of the equation by the LCD, we may have introduced extraneous solutions. An extraneous solution is a solution that is not actually a solution to the original equation.

To check for extraneous solutions, we need to plug each solution back into the original equation and check if it is true.

Example: Checking for Extraneous Solutions

Consider the solutions:

${ x = \frac{3 + \sqrt{37}}{2} }$ ${ x = \frac{3 - \sqrt{37}}{2} }$

We can plug each solution back into the original equation:

${ \frac{3}{x^2+5x+6} + \frac{x-1}{x+2} = \frac{7}{x+3} }$

We can plug in the first solution:

${ \frac{3}{\left( \frac{3 + \sqrt{37}}{2} \right)^2+5\left( \frac{3 + \sqrt{37}}{2} \right)+6} + \frac{\left( \frac{3 + \sqrt{37}}{2} \right)-1}{\left( \frac{3 + \sqrt{37}}{2} \right)+2} = \frac{7}{\left( \frac{3 + \sqrt{37}}{2} \right)+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37}{4}+\frac{15+5\sqrt{37}}{2}+6} + \frac{\frac{3 + \sqrt{37}}{2}-1}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37+30+15\sqrt{37}+72}{4}} + \frac{\frac{3 + \sqrt{37}}{2}-\frac{2}{2}}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37+30+15\sqrt{37}+72}{4}} + \frac{\frac{3 + \sqrt{37}-2}{2}}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37+30+15\sqrt{37}+72}{4}} + \frac{\frac{1 + \sqrt{37}}{2}}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37+30+15\sqrt{37}+72}{4}} + \frac{\frac{1 + \sqrt{37}}{2}}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37+30+15\sqrt{37}+72}{4}} + \frac{\frac{1 + \sqrt{37}}{2}}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37+30+15\sqrt{37}+72}{4}} + \frac{\frac{1 + \sqrt{37}}{2}}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

${ \frac{3}{\frac{9+6\sqrt{37}+37+30+15\sqrt{37}+72}{4}} + \frac{\frac{1 + \sqrt{37}}{2}}{\frac{3 + \sqrt{37}}{2}+2} = \frac{7}{\frac{3 + \sqrt{37}}{2}+3} }$

This simplifies to:

Q&A: Solving Rational Equations

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more fractions with polynomials in the numerator and denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to find the least common denominator (LCD) of the fractions and multiply both sides of the equation by the LCD.

Q: What is the least common denominator (LCD)?

A: The least common denominator (LCD) is the smallest polynomial that is a multiple of all the denominators in the equation.

Q: How do I find the LCD?

A: To find the LCD, you need to factor each denominator and identify the common factors.

Q: What happens when I multiply both sides of the equation by the LCD?

A: When you multiply both sides of the equation by the LCD, you eliminate the fractions and get a polynomial equation that you can solve.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug each solution back into the original equation and check if it is true.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not actually a solution to the original equation.

Q: Why do I need to check for extraneous solutions?

A: You need to check for extraneous solutions because when you multiplied both sides of the equation by the LCD, you may have introduced extraneous solutions.

Q: How do I know if a solution is extraneous or not?

A: You can check if a solution is extraneous or not by plugging it back into the original equation and checking if it is true.

Q: What if I get a solution that is not in the original equation?

A: If you get a solution that is not in the original equation, it is an extraneous solution and you should not include it in your final answer.

Q: Can I use other methods to solve rational equations?

A: Yes, you can use other methods to solve rational equations, such as factoring or using the quadratic formula.

Q: What are some common mistakes to avoid when solving rational equations?

A: Some common mistakes to avoid when solving rational equations include:

• Not finding the LCD
• Not multiplying both sides of the equation by the LCD
• Not checking for extraneous solutions
• Not plugging solutions back into the original equation

Q: How can I practice solving rational equations?

A: You can practice solving rational equations by working through examples and exercises in your textbook or online resources.

Q: What are some real-world applications of solving rational equations?

A: Solving rational equations has many real-world applications, such as:

• Physics: solving equations to describe the motion of objects
• Engineering: solving equations to design and optimize systems
• Economics: solving equations to model and analyze economic systems

Conclusion

Solving rational equations can be challenging, but with practice and patience, you can master this skill. Remember to find the LCD, multiply both sides of the equation by the LCD, and check for extraneous solutions. With these steps, you can solve rational equations and apply them to real-world problems.