Problem 127 - Introductory and Intermediate Algebra for College Students 4th

Question

$$\text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x}$$

Step-by-Step Solution

Answer

The solution is \(x = 3\) and \(x = -1\).

Step 1: Cross Multiplication

Cross multiply the fractions to eliminate the denominator, which gives: \(x(x+2) = 4x + 3\).

Step 2: Expansion

Expand the left-hand side, resulting in \(x^2 + 2x = 4x + 3\).

Step 3: Rearrange

Rearrange to set the equation equal to zero, resulting in \(x^2 + 2x - 4x - 3 = 0\). Simplify to \(x^2 - 2x - 3 = 0\).

Step 4: Factor the Quadratic

Factorize the equation as \((x - 3)(x + 1) = 0\).

Step 5: 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.

Key Concepts

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 \(x(x+2) = 4x + 3\).

• 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.

Cross multiplication is not just limited to rational equations like the one above. It can be applied in various mathematical contexts where two fractions are present.

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.

• 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.

Recognizing a quadratic equation helps determine the appropriate method for solving it.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler factors. It’s 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.

• 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 \(x = -1\).

Factoring is often the quickest and simplest method for solving quadratic equations, provided it is possible and straightforward.

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 \(x^2 - 2x - 3 = 0\).

• 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.

Skilled rearranging can significantly simplify your problem-solving path, making the solving process more apparent and straightforward.