Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 12: Problem 127
$$\text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x}$$
Short Answer
Expert verified
The solution is \(x = 3\) and \(x = -1\).
Step by step solution
01
Cross Multiplication
Cross multiply the fractions to eliminate the denominator, which gives: \(x(x+2) = 4x + 3\).
02
Expansion
Expand the left-hand side, resulting in \(x^2 + 2x = 4x + 3\).
03
Rearrange
Rearrange to set the equation equal to zero, resulting in \(x^2 + 2x - 4x - 3 = 0\). Simplify to \(x^2 - 2x - 3 = 0\).
04
Factor the Quadratic
Factorize the equation as \((x - 3)(x + 1) = 0\).
05
Solve for 'x'
Since the product of two factors is zero, at least one of them must be zero. This gives us two solutions: \(x - 3 = 0\) and \(x + 1 = 0\), giving \(x = 3\) and \(x = -1\) respectively.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross Multiplication
Cross multiplication is a technique often used to solve equations involving two fractions set equal to each other. By cross-multiplying, you eliminate the fractions, resulting in a simpler equation. In this process, the numerator of one fraction is multiplied by the denominator of the other, and vice versa.
For example, in the equation \(\frac{x+2}{4x+3} = \frac{1}{x}\), cross multiplying gives \(x(x+2) = 4x + 3\).
For example, in the equation \(\frac{x+2}{4x+3} = \frac{1}{x}\), cross multiplying gives \(x(x+2) = 4x + 3\).
- This operation clears the denominators, allowing you to focus solely on a straightforward polynomial equation.
- Notice that this is a crucial step because it sets the stage for further manipulation, like expanding and rearranging terms.
Quadratic Equations
Once cross multiplication has been applied, you may find yourself dealing with a quadratic equation. A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown.
In the exercise, after expanding and rearranging terms, the equation \(x^2 - 2x - 3 = 0\) is revealed. This is a textbook quadratic equation.
In the exercise, after expanding and rearranging terms, the equation \(x^2 - 2x - 3 = 0\) is revealed. This is a textbook quadratic equation.
- Quadratic equations are significant because they appear frequently in various areas of math and science.
- To solve these equations, you can use several methods such as factoring, completing the square, or applying the quadratic formula.
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its simpler factors. It鈥檚 a key method for solving quadratic equations when they can be easily factorized.
In the quadratic equation \(x^2 - 2x - 3 = 0\), we factor the expression into \((x - 3)(x + 1) = 0\). This step is vital because it breaks down the equation into simpler components, each of which can be solved separately.
In the quadratic equation \(x^2 - 2x - 3 = 0\), we factor the expression into \((x - 3)(x + 1) = 0\). This step is vital because it breaks down the equation into simpler components, each of which can be solved separately.
- Factoring involves finding the roots, or values of \(x\), that make the equation true by setting each factor equal to zero.
- In our equation, the factors are \(x - 3\) and \(x + 1\). Setting these equal to zero gives \(x = 3\) and \(x = -1\).
Rearranging Equations
Rearranging equations is a common practice in algebra that involves moving terms around to achieve a particular form - often to set the equation to zero or isolate a variable.
In solving \(x^2 + 2x = 4x + 3\), rearranging steps included bringing all terms to one side, resulting in \(x^2 + 2x - 4x - 3 = 0\). This simplifies further to \(x^2 - 2x - 3 = 0\).
In solving \(x^2 + 2x = 4x + 3\), rearranging steps included bringing all terms to one side, resulting in \(x^2 + 2x - 4x - 3 = 0\). This simplifies further to \(x^2 - 2x - 3 = 0\).
- Rearranging makes the equation easier to solve by combining like terms and simplifying the expression.
- This step prepares the quadratic equation for either factoring, using the quadratic formula, or another solving method.
