Problem 49 - College Algebra

Question

Exercises \(31-50\) contain 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. $$ \frac{1}{x-4}-\frac{5}{x+2}=\frac{6}{x^{2}-2 x-8} $$

Step-by-Step Solution

Answer

The restrictions on \(x\) are 4 from \(x-4 = 0\), and -2 from \(x + 2 = 0\). Then clear the fractions and solve the result quadratic equation. Check the solutions against the restrictions. Discard any solutions that are also restrictions as they are extraneous.

Step 1: Identify the restrictions

The restrictions are the values that would make the denominator zero. These are found by setting \(x-4 = 0\) and \(x+2 = 0\), and solving for \(x\). Next, as the denominator of the third fraction in question is a quadratic equation, set \(x^{2}-2x-8 = 0\), factor it and also solve it for \(x\). These are the restrictions on \(x\) and must be excluded from our solution set.

Step 2: Clear the fractions

Multiply each term in the equation by the common denominator, in this case the denominator is \(x^{2}-2 x-8\). That will clear the fractions and will turn the equation into a quadratic equation.

Step 3: Solve the equation

After the equation has been transformed to a quadratic equation, solve it by factoring, completing the square, or applying the quadratic formula, whichever is most suitable.

Step 4: Check the solution against the restrictions

No solution can be a value that makes the denominator of the initial equation zero. After getting the solutions from the quadratic equation, make sure that none of the solutions are equal to the restrictions found previously. If a solution is a restriction, then that solution is extraneous and should be discarded.

Key Concepts

Denominators

In rational equations, denominators play a crucial role for several reasons. A denominator in a fraction cannot be zero because division by zero is undefined in mathematics. Therefore, it's essential to determine any values that make a denominator zero.
When looking at a rational equation, identify each unique denominator in the equation. In our exercise, we had denominators:

• \(x - 4\)
• \(x + 2\)
• \(x^2 - 2x - 8\)

Knowing these denominators helps us find restrictions that will guide us through solving the equation correctly.

Restrictions

Restrictions in rational equations are values that make one or more denominators equal to zero, and hence must be excluded from possible solutions.
To find these restrictions, set each denominator to zero and solve for \(x\):

• For \(x-4=0\), \(x = 4\)
• For \(x+2=0\), \(x = -2\)

For the quadratic denominator \(x^2 - 2x - 8 = 0\), solve it by:

• Factoring to \((x-4)(x+2) = 0\)
• This gives the same restrictions \(x=4\) and \(x=-2\)

These are the restrictions, and no solutions of the equation should match these values, as those solutions would make the original equation undefined.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), and they are key to solving some rational equations after clearing out denominators.
In our exercise, once the fractions were cleared by multiplying through by the common denominator, we transformed the rational equation into a quadratic equation. Identifying this transformed equation correctly is crucial as it will guide how we approach solving for \(x\). Understanding the basic properties of quadratics, such as their standard form and how they graph as parabolas, aids in interpreting solutions.

Factoring

Factoring is a method to solve quadratic equations when the quadratic can be expressed as a product of its binomials. Factoring involves finding two numbers that multiply to give the constant term \(c\) and add to give the middle coefficient \(b\).
For example, to factor \(x^2 - 2x - 8\):

• Look for two numbers that multiply to \(-8\) and add to \(-2\)
• These numbers are \(-4\) and \(+2\)
• So, \(x^2 - 2x - 8\) factors to \((x-4)(x+2)\)

This technique simplifies solving the quadratic by setting each factor equal to zero, which helps find the roots of the equation.

Quadratic Formula

The quadratic formula is a universal method for finding solutions to any quadratic equation, and is invaluable when factoring is complicated or not feasible. It's given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In scenarios where either the equation is difficult to factor, or for confirmation of manually factored solutions, apply the quadratic formula. It ensures you consider all possible roots of the equation. Remember to check any solutions against previously determined restrictions to ensure they don’t render the original equation undefined.