Problem 53 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation. $$ \frac{3 x}{x-2}+\frac{4}{x+2}=\frac{24}{x^{2}-4} $$

Step-by-Step Solution

Answer

The solutions for the given equation are \(x = \frac{-10 + \sqrt{424}}{6}\) or \(x = \frac{-10 - \sqrt{424}}{6}\).

Step 1: Identify the denominators

We first notice that the denominators of the given fractions are (x-2), (x+2), and \((x^2-4)\). The last denominator is the difference of squares, so it can be factored as (x-2)(x+2). The given equation is: \[ \frac{3 x}{x-2}+\frac{4}{x+2}=\frac{24}{x^{2}-4} \]

Step 2: Find the LCD

The LCD of the denominators is the product of unique factors: \[ LCD = (x-2)(x+2) \]

Step 3: Rewrite the fractions with the LCD

Now, we rewrite each fraction such that it has the LCD as its denominator. In order to do this, we need to multiply each fraction by the appropriate factor, so that the new denominators match the LCD. Here's what we get: \[ \frac{3 x(x+2)}{(x-2)(x+2)}+\frac{4(x-2)}{(x-2)(x+2)}=\frac{24}{(x-2)(x+2)} \]

Step 4: Simplify and solve for x

Now, we can multiply both sides of the equation by the LCD to eliminate the denominators: \[ 3x(x+2) + 4(x-2) = 24 \] Next, we distribute and simplify the equation: \[ 3x^2+6x+4x-8=24 \] Combining like terms, we get: \[ 3x^2+10x-32=0 \] Now, let's solve this quadratic equation for x either by factoring or by applying the quadratic formula. Since factoring is a bit difficult in this case, we will use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \] Substitute the values of a, b, and c from our quadratic equation, 3x^2+10x-32=0: \[ x = \frac{-10 \pm \sqrt{10^2-4(3)(-32)}}{2(3)} \] Calculate the values within the square root and simplify the expression: \[ x = \frac{-10 \pm \sqrt{424}}{6} \] Now, we have two possible solutions for x: \[ x = \frac{-10 + \sqrt{424}}{6} \textrm{ or } x = \frac{-10 - \sqrt{424}}{6} \] As we can see, the solutions for x are not whole numbers or easily simplified, but these two expressions give us the correct solutions for x.

Key Concepts

Least Common Denominator (LCD)

When solving an equation that involves multiple fractions, finding the least common denominator (LCD) is a crucial step to simplifying the problem. The LCD is the smallest number that each of the denominators can divide into without leaving a remainder. In other words, it's the smallest common multiple of all denominators involved.

In the case of our exercise, the denominators are \(x-2\), \(x+2\), and \(x^2-4\). To find the LCD, we examine the factors of each denominator. Here, \(x^2-4\) is a 'difference of squares' which factors into \(x-2\) and \(x+2\). Therefore, the LCD for all three fractions is simply \(x-2\) times \(x+2\).

Finding the LCD allows us to combine the fractions on a common denominator, setting the stage for easier simplification and ultimately solving for the variable.

Difference of Squares

The 'difference of squares' is a mathematical term that describes a binomial of the form \(a^2 - b^2\). This binomial can always be factored into \(a+b\) times \(a-b\). For instance, in our exercise, we encounter the term \(x^2-4\) which is a difference of squares because it can be written as \(x^2-2^2\).

According to the difference of squares formula, \(x^2-4\) factors into \(x+2\) and \(x-2\). Recognizing this pattern is incredibly useful when solving equations involving fractions, as it simplifies finding the LCD and reduces the complexity involved when combining fractions or moving towards the solution of the equation.

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree, which means it contains at least one term that is squared. The standard form of a quadratic equation is \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\).

In the process of solving our equation, after eliminating the denominators and combining like terms, we obtain a quadratic equation: \(3x^2+10x-32=0\). Solving this equation for \(x\) involves either factoring, completing the square, or applying the quadratic formula. As our equation doesn’t easily factor, the quadratic formula offers a more efficient route to finding the values of \(x\) that satisfy the equation.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions to quadratic equations. It is stated as \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the numerical coefficients taken from the standard form of the quadratic equation \(ax^2+bx+c=0\). The symbol \(\pm\) indicates that there will be two solutions.

In our exercise, we substitute \(a=3\), \(b=10\), and \(c=-32\) into the quadratic formula, and perform the arithmetic to find the solutions for \(x\). This formula provides a method for solving quadratic equations when factoring is not possible or practical, ensuring that a solution can always be found algebraically.