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Chapter 6: Problem 72

Solve each equation, and check the solutions. $$ \frac{-13}{t^{2}+6 t+8}+\frac{4}{t+2}=\frac{3}{2 t+8} $$

Short Answer

Expert verified
The solution is \(t = 0\).

Step by step solution

01

Factor the Quadratic Expression

Factor the quadratic expression in the denominator of the first fraction: \[ t^2 + 6t + 8 = (t + 2)(t + 4) \] This helps us break down the problem.
02

Re-write the Equation with Factored Denominators

Rewrite the equation with the factored form: \[ \frac{-13}{(t + 2)(t + 4)} + \frac{4}{t + 2} = \frac{3}{2(t + 4)} \] Now, the denominators show their common factors.
03

Combine Terms Over a Common Denominator

Identify the common denominator for all terms, which is \[2(t+2)(t+4)\]. Rewrite each fraction with this common denominator: \[ \frac{-13}{(t + 2)(t + 4)} = \frac{-13 \times 2}{2(t+2)(t+4)} \ \frac{4}{t + 2} = \frac{4 \times 2(t+4)}{2(t+2)(t+4)} \ \frac{3}{2(t + 4)} = \frac{3(t+2)}{2(t+2)(t+4)} \] Now, combine these terms under one common denominator.
04

Simplify and Combine Fractions

Combine the fractions under the common denominator: \[ \frac{-26 + 8(t+4) - 3(t+2)}{2(t+2)(t+4)} \] Simplify the numerator by distributing and combining like terms.
05

Solve the Simplified Equation

Simplify the numerator: \[ -26 + 8t + 32 - 3t - 6 = 5t \] This results in the simplified equation: \[ \frac{5t}{2(t+2)(t+4)} = 0 \] The only way for this fraction to equal zero is if the numerator equals zero: \[ 5t = 0 \]. Solve for \(t\): \[ t = 0 \]
06

Check the Solution

Check by substituting \(t = 0\) back into the original equation. Substitute and compute: \[ \frac{-13}{0^2 + 6 \times 0 + 8} + \frac{4}{0 + 2} = \frac{3}{2 \times 0 + 8} \] This simplifies to: \[ \frac{-13}{8} + 2 = \frac{3}{8} \] Continue simplifying: \[ -\frac{13}{8} + \frac{16}{8} = \frac{3}{8} \] This confirms the solution is correct.

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Key Concepts

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

Solving Rational Equations
Rational equations involve fractions with polynomials in their numerator and/or denominator. To solve these equations, we follow a methodical approach:

- Identify the least common denominator (LCD) for all fractions.
- Rewrite each fraction with the LCD as its denominator.
- Combine the fractions into a single rational expression.
- Solve the resulting equation.

For our exercise, we first factored the quadratic expression in the denominator, then identified the common denominator as 2(t+2)(t+4). This allowed us to combine the fractions and ultimately solve for the variable.
Factoring Quadratics
Factoring quadratics is essential for simplifying rational expressions and solving equations. A quadratic expression is of the form \(ax^2 + bx + c\). Steps to factor a quadratic:

- Find two numbers that multiply to give the product of \(a \times c\) and add to give the coefficient b.
- Rewrite the middle term using these two numbers.
- Factor by grouping.

In our exercise, we factored \(t^2 + 6t + 8\) into \((t + 2)(t + 4)\). This made it easier to find a common denominator and simplify the rational equation.
Common Denominators
Finding a common denominator is crucial in solving rational equations. The common denominator is a shared multiple of all denominators involved. Steps to find and use a common denominator:

- Factor all denominators completely.
- Identify the LCD by taking the highest power of each factor.
- Rewrite each fraction with this LCD.
- Combine the fractions into one and simplify.

In our problem, the common denominator for the terms was \(2(t+2)(t+4)\). By rewriting each fraction with this common denominator, we could combine and simplify the fractions effectively to solve for the variable.

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Most popular questions from this chapter

Solve each problem. The percent of deaths caused by smoking is modeled by the rational expression $$\frac{x-1}{x}$$ where \(x\) is the number of times a smoker is more likely than a nonsmoker to die of lung cancer. This is called the incidence rate. (Data from Walker, A., Observation and Inference: An Introduction to the Methods of Epidemiology, Epidemiology 91影视 Inc.) For example, \(x=10\) means that a smoker is 10 times more likely than a nonsmoker to die of lung cancer. Find the percent of deaths if the incidence rate is the given number. (a) 5 (b) 10 (c) 20 (d) Can the incidence rate equal \(0 ?\) Explain.

Solve each formula or equation for the specified variable. $$ I=\frac{k E}{n} \text { for } E $$

Solve each formula or equation for the specified variable. $$ -3 t-\frac{4}{p}=\frac{6}{s} \text { for } p $$

Solve each formula or equation for the specified variable. $$ m=\frac{k F}{a} \text { for } F $$

Find the slope of the line that passes through each pair of points. This will involve simplifying complex fractions. $$ \left(-\frac{2}{9}, \frac{5}{18}\right) \text { and }\left(\frac{1}{18},-\frac{5}{9}\right) $$
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