Problem 46 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$\frac{x^{2}}{x+9}-\frac{11}{x+9}=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \sqrt{11}\) and \(x = -\sqrt{11}\).

Step 1: Combine fractions with common denominator

The fractions have common denominators, so they can be combined. This results in \(\frac{x^2 - 11}{x+9} = 0\).

Step 2: Solve for x

The fraction is equal to zero, which means that either the numerator or the denominator is zero. If the denominator is zero, then there's no solution. In this case the denominator is not zero, so the numerator must be zero. Therefore, \(x^2 -11 = 0\). Solve this quadratic equation to find the value of \(x\).

Step 3: Solve quadratic equation

The quadratic can be solved using the following steps: Add 11 to both sides of the equation to get \(x^2 = 11\). Then, take square root on both sides, we get \(x = \sqrt{11}\) and \(x = -\sqrt{11}\).

Step 4: Validate the solutions

Make sure that the solutions don't make the denominator equal to zero in the original equation. For \(\sqrt{11}\), use \(x = \sqrt{11}\) in \(x+9\). The result would be \(\sqrt{11} + 9\), which is not equal to zero. Similarly, for \(-\sqrt{11}\), use \(x = -\sqrt{11}\) in \(x+9\). The result would be \(-\sqrt{11} + 9\), which is also not zero. Hence, both solutions are validated.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x with the standard form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. While these equations can sometimes appear complex, there are various methods to solve them, such as factoring, completing the square, using the quadratic formula, or graphing.

For instance, the exercise provided \( \frac{x^{2}}{x+9}-\frac{11}{x+9}=0 \) simplifies to a quadratic equation once a common denominator is utilized. The equation can be rewritten as \( x^2 - 11 = 0 \) after combining like terms. Solving this quadratic equation leads to the solutions \( x = \sqrt{11} \) and \( x = -\sqrt{11} \).

Common Denominators

When dealing with fractions, the term 'common denominator' refers to a shared denominator that allows us to perform operations like addition, subtraction, and comparison between fractions. Finding a common denominator is a crucial step in simplifying complex expressions and solving equations that involve fractions.

In our example, the fractions \( \frac{x^2}{x+9} \) and \( \frac{11}{x+9} \) share the same denominator, \( x+9 \). This allows us to combine them into a single fraction. Not having to deal with multiple denominators simplifies the problem and leads us closer to finding the solution for x.

Simplify Irrational Solutions

Irrational solutions are roots that cannot be expressed as a simple fraction. When solving quadratic equations, we often encounter square roots that result in such solutions. Simplifying these roots may not always result in a rational number, but it involves writing them in the simplest radical form possible.

In the solution for the given exercise, we have \( \sqrt{11} \) and \( -\sqrt{11} \) which are already in the simplest form. You cannot simplify \( \sqrt{11} \) further since 11 is not a perfect square. Therefore, in cases like these, the irrational solutions are presented as is.

Validate Solutions

Validating solutions is an essential step in solving equations. It ensures that the solutions found do not result in undefined expressions, such as division by zero, when substituted back into the original equation.

In the context of our example, we check whether the values \( \sqrt{11} \) and \( -\sqrt{11} \) make the denominator of the original expression \( x+9 \) equal to zero. After substituting these back into the denominator, we see they do not lead to zero, which means our solutions are valid. It's a crucial final check that prevents incorrect solutions from being accepted.