Problem 50 - Algebra and Trigonometry

Question

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{6}{x+3}-\frac{5}{x-2}=\frac{-20}{x^{2}+x-6}$$

Step-by-Step Solution

Answer

The restrictions on \(x\) are \(x\neq-3\) and \(x\neq2\). The quadratic equation \(x^{2}-4x+12 = 0\) is obtained when the equation is simplified. The solutions to the quadratic equation must be tested against the restrictions to confirm their validity.

Step 1: Identify restrictions on variables

Firstly, the denominators in this equation are \(x+3\), \(x-2\) and \(x^{2}+x-6\). Setting each of these equal to zero and solving for \(x\) gives the following restrictions on \(x\): \(x = -3\), \(x = 2\) and \(x= -3, 2\), respectively.

Step 2: Simplify the equation

Next, simplify the equation by clearing the fractions. This can be done by multiplying each term of the equation by the lowest common multiple (LCM) of the denominators. The LCM is \(x^{2}+x-6\). So, the equation becomes: \[6(x^{2}+x-6) - 5(x^{2}+x-6) = -20.\] Simplifying gives \[x^{2}-4x+12 = 0.\]

Step 3: Solve the simplified equation

Now, solve the simplified quadratic equation \(x^{2}-4x+12 = 0\). Using the quadratic formula \(x=(-b ± sqrt{b^{2}-4ac})/2a\) gives solutions for \(x\). However, these solutions must not be any of the restrictions identified in Step 1.

Step 4: Test Solutions against restrictions

Check the solutions found against the restrictions on \(x\) identified in Step 1. In this case, possible solutions for \(x\) must not be -3, 2. If they are, they must be rejected as not valid.

Key Concepts

Variables in Denominators

In rational equations, we often encounter variables in the denominators. This presence of variables can make equations more complex than simple numeric denominators because they introduce conditions for equality. For example, consider the rational equation \(\frac{6}{x+3}-\frac{5}{x-2}=\frac{-20}{x^{2}+x-6}\). Here, the denominators \(x+3\), \(x-2\), and \(x^{2}+x-6\) consist of variables, which affect how we approach solving the equation.

Equations with variable denominators require careful attention because the values of \(x\) can lead to undefined expressions. For this reason, steps are needed to ensure that when we modify or solve the equation, the variable's range remains valid or unaffected by restrictions that lead to division by zero.

Restrictions on Variables

When solving rational equations with variables in the denominator, identifying restrictions is a necessary first step. Restrictions occur at values where any denominator equals zero, as division by zero is undefined in mathematics.

For this exercise, we first set each denominator equal to zero:

• \(x+3 = 0\) results in \(x = -3\).
• \(x-2 = 0\) results in \(x = 2\).
• Solving \(x^{2}+x-6 = 0\) using factorization or the quadratic formula, confirms restrictions at \(x = -3\) and \(x = 2\).

These values highlight where the equation is undefined, so they must not be part of our solution set. Recognizing these restrictions before proceeding ensures that the solutions we find will be valid.

Quadratic Equation

After addressing restrictions, rational equations often require simplification into a more manageable form. In our example, eliminating variables from the denominators transforms the rational equation into a quadratic equation, \(x^{2}-4x+12=0\).

Quadratic equations maintain a standard form expressed as \(ax^{2} + bx + c = 0\). Solutions can often be found using the quadratic formula, which is versatile enough for all quadratic types:
\[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\\] Here, "solving" implies finding the roots of the equation, which are the \(x\)-values satisfying the equation. It's important to remember that these solutions must respect any initial restrictions identified to avoid invalid results.

Solving Equations

Solving equations, especially those like the one we're discussing, requires a structured approach. Here, we focus on deriving solutions while adhering strictly to the restrictions previously found.

Using the quadratic formula on \(x^{2}-4x+12=0\), we calculate the possible values of \(x\). Let's say after computation, we find potential solutions such as \(x = a\) and \(x = b\). It's crucial to check these against our restrictions: \(x eq -3\) and \(x eq 2\). If either solution aligns with these restricted values, it would be discarded.

The final stage of solving includes verification, a key step to validate that substituted values do satisfy the original equations. In strategic problem-solving, not only do we find potential solutions, but we ensure they work within established mathematical limits.