Solve For $x$: 2 X 2 X 2 − 9 = 2 X − 3 + 3 X X + 3 \frac{2 X^2}{x^2-9}=\frac{2}{x-3}+\frac{3 X}{x+3} X 2 − 9 2 X 2 ​ = X − 3 2 ​ + X + 3 3 X ​ If There Is More Than One Solution, Separate Them. If There Is No Solution, State No Solution.

by ADMIN 242 views

Introduction

Solving rational equations can be a challenging task, especially when dealing with complex fractions. In this article, we will focus on solving the equation 2x2x2−9=2x−3+3xx+3\frac{2 x^2}{x^2-9}=\frac{2}{x-3}+\frac{3 x}{x+3}. We will break down the solution step by step, using algebraic manipulations and factoring techniques to isolate the variable xx.

Step 1: Factor the Denominators

The first step in solving this equation is to factor the denominators of the fractions. We can factor the denominator of the left-hand side as (x−3)(x+3)(x-3)(x+3), and the denominators of the right-hand side as (x−3)(x-3) and (x+3)(x+3), respectively.

$\frac{2 x^2}{(x-3)(x+3)}=\frac{2}{x-3}+\frac{3 x}{x+3}$

Step 2: Multiply Both Sides by the Least Common Denominator

To eliminate the fractions, we need to multiply both sides of the equation by the least common denominator (LCD), which is (x−3)(x+3)(x-3)(x+3). This will allow us to work with polynomials instead of fractions.

$2 x^2 = 2(x+3) + 3x(x-3)$

Step 3: Expand and Simplify

Next, we need to expand and simplify the right-hand side of the equation. We can do this by multiplying the terms inside the parentheses and combining like terms.

$2 x^2 = 2x + 6 + 3x^2 - 9x$

Step 4: Combine Like Terms

Now, we can combine like terms on the right-hand side of the equation. We can add or subtract the coefficients of the same variables.

$2 x^2 = 3x^2 - 7x + 6$

Step 5: Move All Terms to One Side

To isolate the variable xx, we need to move all the terms to one side of the equation. We can do this by subtracting 3x23x^2 from both sides and adding 7x7x to both sides.

$0 = x^2 - 7x + 6$

Step 6: Factor the Quadratic Expression

The next step is to factor the quadratic expression on the right-hand side of the equation. We can factor the expression as (x−3)(x−2)(x-3)(x-2).

$0 = (x-3)(x-2)$

Step 7: Solve for xx

Now that we have factored the quadratic expression, we can solve for xx by setting each factor equal to zero. We can set x−3=0x-3=0 and x−2=0x-2=0 and solve for xx.

$x-3=0 \Rightarrow x=3$
$x-2=0 \Rightarrow x=2$

Conclusion

In conclusion, we have solved the equation 2x2x2−9=2x−3+3xx+3\frac{2 x^2}{x^2-9}=\frac{2}{x-3}+\frac{3 x}{x+3} and found two solutions: x=3x=3 and x=2x=2. These solutions satisfy the original equation and can be verified by plugging them back into the equation.

Discussion

The equation 2x2x2−9=2x−3+3xx+3\frac{2 x^2}{x^2-9}=\frac{2}{x-3}+\frac{3 x}{x+3} is a rational equation that involves complex fractions. To solve this equation, we need to use algebraic manipulations and factoring techniques to isolate the variable xx. The solutions to this equation are x=3x=3 and x=2x=2, which can be verified by plugging them back into the equation.

Final Answer

The final answer is: 3,2\boxed{3, 2}

Introduction

Solving rational equations can be a challenging task, especially when dealing with complex fractions. In this article, we will provide a Q&A section to help you better understand the process of solving rational equations. We will cover common questions and provide step-by-step solutions to help you master this important math concept.

Q1: What is a rational equation?

A rational equation is an equation that contains fractions, where the numerator and denominator are polynomials. Rational equations can be solved using algebraic manipulations and factoring techniques.

Q2: How do I solve a rational equation?

To solve a rational equation, you need to follow these steps:

  1. Factor the denominators of the fractions.
  2. Multiply both sides of the equation by the least common denominator (LCD).
  3. Expand and simplify the right-hand side of the equation.
  4. Combine like terms on the right-hand side of the equation.
  5. Move all terms to one side of the equation.
  6. Factor the quadratic expression on the right-hand side of the equation.
  7. Solve for x by setting each factor equal to zero.

Q3: What is the least common denominator (LCD)?

The least common denominator (LCD) is the smallest expression that can be divided by both denominators of the fractions in the equation. To find the LCD, you need to factor the denominators and find the smallest expression that contains all the factors.

Q4: How do I find the LCD?

To find the LCD, you need to factor the denominators and find the smallest expression that contains all the factors. For example, if the denominators are (x-3) and (x+3), the LCD is (x-3)(x+3).

Q5: What is the difference between a rational equation and a rational expression?

A rational equation is an equation that contains fractions, while a rational expression is an expression that contains fractions. Rational expressions can be simplified or manipulated using algebraic techniques, but they are not necessarily equal to zero.

Q6: Can I use a calculator to solve rational equations?

Yes, you can use a calculator to solve rational equations. However, it's always a good idea to check your work by plugging the solution back into the original equation.

Q7: What are some common mistakes to avoid when solving rational equations?

Some common mistakes to avoid when solving rational equations include:

  • Not factoring the denominators correctly
  • Not multiplying both sides of the equation by the LCD
  • Not expanding and simplifying the right-hand side of the equation correctly
  • Not combining like terms correctly
  • Not factoring the quadratic expression correctly

Q8: Can I use algebraic manipulations to simplify rational expressions?

Yes, you can use algebraic manipulations to simplify rational expressions. For example, you can multiply both the numerator and denominator by the same expression to simplify the fraction.

Q9: What is the importance of solving rational equations?

Solving rational equations is an important skill in mathematics, as it allows you to solve equations that contain fractions. Rational equations are used in a variety of applications, including physics, engineering, and economics.

Q10: Can I use rational equations to model real-world problems?

Yes, you can use rational equations to model real-world problems. For example, you can use rational equations to model the motion of an object, the growth of a population, or the flow of a fluid.

Conclusion

In conclusion, solving rational equations is an important skill in mathematics that requires algebraic manipulations and factoring techniques. By following the steps outlined in this article, you can master the process of solving rational equations and apply it to a variety of real-world problems.

Final Answer

The final answer is: There is no final answer, as this article is a Q&A section.