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Chapter 6: Problem 48

Solve each equation. $$\frac{1}{x+4}+\frac{1}{3}=\frac{-10}{3\left(x^{2}-16\right)}$$

Short Answer

Expert verified
x = 3

Step by step solution

01

- Identify the common denominator

Notice that the equation contains fractions. A useful first step is to identify a common denominator for all terms. In this case, the common denominator is \ \(3(x+4)(x-4)\ \) or equivalently \ \(3 \times (x^2 - 16)\ \).
02

- Multiply each term by the common denominator

Multiply every term in the equation by the common denominator \ \(3(x+4)(x-4)\ \). This eliminates the denominators: \ \[3(x+4)(x-4) \times \frac{1}{x+4} + 3(x+4)(x-4) \times \frac{1}{3} = 3(x+4)(x-4) \times \frac{-10}{3(x^{2}-16)}\ \]
03

- Simplify each term

Simplify the equation: \ \[3(x-4) + (x+4)(x-4) = -10\ \] Continue simplifying: \ \[3(x-4) + (x^2-16) = -10\ \]
04

- Combine like terms

Combine and simplify the expression: \ \[3x - 12 + x^2 - 16 = -10\ \] Which simplifies to \ \[x^2 + 3x - 28 = -10\ \] Move all terms to one side: \ \[x^2 + 3x - 18 = 0\ \]
05

- Solve the quadratic equation

Solve the quadratic equation \ \[x^2 + 3x - 18 = 0\ \] To solve for \ \ x\ \ , either factorize, use the quadratic formula \ \[x = \frac{{-b \, \, \, \, \, \, \, \, \pm \, \, \, \, \, \, \sqrt{{b^2 - 4ac}}}}{{2a}}\ \] In this context, factoring gives us: \ \[(x+6)(x-3) = 0\ \] Thus, \ \ x = -6\ \ or \ \ x = 3\ \
06

- Verify solutions

Substitute x = -6 and x = 3 back into the original equation to ensure they do not make any denominators zero. Only x = 3 is a valid solution since x = -6 leads to division by zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When dealing with fractions in an equation, finding a common denominator simplifies the process. For example, consider the equation \(\frac{1}{x+4}+\frac{1}{3}=\frac{-10}{3(x^{2}-16)} \). The key is to determine the smallest common multiple of all denominators. Here, the common denominator is \[ 3(x+4)(x-4) \] or equivalently \[ 3 \times (x^2 - 16) \]. This transformation helps combine and eliminate the fractions, making the equation easier to solve.
Quadratic Equation
A quadratic equation typically looks like \[ ax^2 + bx + c = 0 \]. In our exercise, after simplifying and moving all terms to one side, the equation becomes \[ x^2 + 3x - 18 = 0 \]. Such equations can be solved using various methods such as factoring, completing the square, or the quadratic formula. Here, we chose factoring.
Factoring
Factoring involves expressing a quadratic equation as a product of its roots. For \[ x^2 + 3x - 18 = 0 \], we need to find two numbers that multiply to -18 and add up to 3. These numbers are 6 and -3. Therefore, the equation can be factored as \[ (x+6)(x-3) = 0 \]. Solving this gives \[ x = -6 \] and \[ x = 3 \].
Simplifying Fractions
Simplifying fractions is a crucial step in solving rational equations. By multiplying both sides of the equation by the common denominator, we eliminate the fractions. For instance, multiplying \[ \frac{1}{x+4} \] by \[ 3(x+4)(x-4) \] results in \[ 3(x-4) \], effectively removing the denominator and simplifying the term. This step makes subsequent algebraic manipulations much easier and the equation more straightforward to solve.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{p^{2}-25}{4 p} \cdot \frac{2}{5-p} $$

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