What Is The Solution To The Following Equation?${ \frac{12}{x^3 + 5x^2 - 4x - 20} = \frac{4}{x^2 + 7x + 10} }$A. No Solution B. X = 5 X = 5 X = 5 C. X = 21 X = 21 X = 21 D. X = 14 X = 14 X = 14

Introduction

In mathematics, solving equations is a fundamental concept that helps us understand various mathematical operations and relationships. When we encounter an equation, our primary goal is to find the value or values of the variable that satisfy the equation. In this article, we will focus on solving a specific equation involving fractions and polynomials.

The Given Equation

The equation we are given is:

$$\frac{12}{x^3 + 5x^2 - 4x - 20} = \frac{4}{x^2 + 7x + 10}$$

Our task is to find the solution to this equation, which means we need to determine the value or values of $x$ that make the equation true.

Step 1: Factor the Denominators

To solve this equation, we first need to factor the denominators of both fractions. Factoring the denominators will help us simplify the equation and make it easier to solve.

The denominator of the first fraction, $x^3 + 5x^2 - 4x - 20$, can be factored as:

$$(x + 5)(x^2 - 4)$$

The denominator of the second fraction, $x^2 + 7x + 10$, can be factored as:

$$(x + 5)(x + 2)$$

Step 2: Cancel Common Factors

Now that we have factored the denominators, we can cancel common factors between the two fractions. The common factor is $(x + 5)$, which appears in both denominators.

Canceling the common factor $(x + 5)$, we get:

$$\frac{12}{(x^2 - 4)} = \frac{4}{(x + 2)}$$

Step 3: Cross-Multiply

To eliminate the fractions, we can cross-multiply the two sides of the equation. Cross-multiplying means multiplying both sides of the equation by the denominators of both fractions.

Cross-multiplying, we get:

$$12(x + 2) = 4(x^2 - 4)$$

Step 4: Expand and Simplify

Now that we have cross-multiplied, we can expand and simplify the equation. Expanding the left-hand side of the equation, we get:

$$12x + 24 = 4x^2 - 16$$

Simplifying the equation, we get:

$$4x^2 - 12x - 40 = 0$$

Step 5: Solve the Quadratic Equation

The equation we have obtained is a quadratic equation, which can be solved using various methods such as factoring, completing the square, or using the quadratic formula.

Using the quadratic formula, we get:

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

where $a = 4$, $b = -12$, and $c = -40$.

Plugging in the values, we get:

$$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(-40)}}{2(4)}$$

Simplifying the expression, we get:

$$x = \frac{12 \pm \sqrt{144 + 640}}{8}$$

$$x = \frac{12 \pm \sqrt{784}}{8}$$

$$x = \frac{12 \pm 28}{8}$$

Step 6: Find the Solutions

Now that we have obtained the solutions using the quadratic formula, we can find the values of $x$ that satisfy the equation.

The two solutions are:

$$x = \frac{12 + 28}{8} = \frac{40}{8} = 5$$

$$x = \frac{12 - 28}{8} = \frac{-16}{8} = -2$$

Conclusion

In conclusion, the solution to the given equation is $x = 5$. This means that when $x = 5$, the equation is satisfied, and the two fractions are equal.

The other options, $x = 21$ and $x = 14$, are not solutions to the equation.

Therefore, the correct answer is:

A. No solution

B. $x = 5$

C. $x = 21$

D. $x = 14$

The final answer is $x = 5$.

Introduction

Solving equations is a fundamental concept in mathematics that helps us understand various mathematical operations and relationships. In our previous article, we discussed how to solve a specific equation involving fractions and polynomials. In this article, we will address some frequently asked questions (FAQs) about solving equations.

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to read and understand the equation. This involves identifying the variable, the constants, and the mathematical operations involved.

Q: How do I simplify an equation?

A: To simplify an equation, you can combine like terms, cancel common factors, and use algebraic properties such as the distributive property.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows us to solve a quadratic equation of the form ax^2 + bx + c = 0. The formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I determine the number of solutions to an equation?

A: To determine the number of solutions to an equation, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two solutions. If the discriminant is zero, the equation has one solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the difference between a solution and a root?

A: A solution to an equation is a value that makes the equation true, while a root of an equation is a value that makes the equation equal to zero.

Q: How do I check if a solution is correct?

A: To check if a solution is correct, you can plug the solution back into the original equation and simplify. If the equation is true, then the solution is correct.

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

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

• Not reading and understanding the equation
• Not simplifying the equation
• Not using algebraic properties such as the distributive property
• Not checking if a solution is correct

Conclusion

Solving equations is a fundamental concept in mathematics that requires practice and patience. By understanding the steps involved in solving equations and avoiding common mistakes, you can become proficient in solving equations and apply this skill to various mathematical problems.

Additional Resources

If you are struggling with solving equations or need additional practice, there are many online resources available, including:

• Khan Academy: A free online platform that offers video lessons and practice exercises on solving equations.
• Mathway: A free online calculator that can help you solve equations and check your work.
• IXL: A subscription-based online platform that offers practice exercises and quizzes on solving equations.

Remember, practice is key to becoming proficient in solving equations. With consistent practice and review, you can develop the skills and confidence needed to solve equations and apply this skill to various mathematical problems.