Problem 98 - Algebra for College Students

Question

Find all real or imaginary solutions to each equation. Use the method of your choice. $$\frac{1}{x}-\frac{2}{1-x}=\frac{1}{2}$$

Step-by-Step Solution

Answer

The solutions are \[x = \frac{7 + \sqrt{57}}{2}\] and \[x = \frac{7 - \sqrt{57}}{2}\].

Step 1: Find a common denominator

To solve the equation \(\frac{1}{x} - \frac{2}{1-x} = \frac{1}{2}\), first find a common denominator for the fractions. The common denominator for \x\ and \1-x\ is \(x(1-x)\).

Step 2: Rewrite each term

Rewrite each term with the common denominator \(x(1-x)\): \(\frac{1}{x} = \frac{1-x}{x(1-x)}\) and \(\frac{2}{1-x} = \frac{2x}{x(1-x)}\). The equation then becomes \(\frac{1-x}{x(1-x)} - \frac{2x}{x(1-x)} = \frac{1}{2}\).

Step 3: Combine the fractions

Combine the fractions on the left side: \(\frac{1-x-2x}{x(1-x)} = \frac{1}{2}\). This simplifies to \(\frac{1-3x}{x(1-x)} = \frac{1}{2}\).

Step 4: Cross multiply to clear the fractions

Multiply both sides by \2x(1-x)\ to clear the fractions: \[2(1-3x) = x(1-x)\].

Step 5: Simplify the equation

Expand and simplify: \[2 - 6x = x - x^2\]. Rearrange to get a quadratic equation: \[-x^2 + 7x - 2 = 0\].

Step 6: Solve the quadratic equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve \(-x^2 + 7x - 2 = 0\). Here, \a = -1\, \b = 7\, and \c = -2\. Plug in the values: \[x = \frac{-7 \pm \sqrt{49 - (-8)}}{2(-1)}\] which simplifies to \[x = \frac{-7 \pm \sqrt{57}}{-2} = \frac{7 \pm \sqrt{57}}{2}\].

Step 7: Determine the solutions

Thus, the solutions to the equation are: \[x = \frac{7 + \sqrt{57}}{2}\] and \[x = \frac{7 - \sqrt{57}}{2}\].

Key Concepts

common denominators

When you come across an equation involving fractions, one of the essential steps is to find a common denominator. A common denominator is a shared multiple of the denominators of the fractions involved. It allows you to combine the fractions into a single fraction, making manipulation easier.
In our equation \( \frac{1}{x} - \frac{2}{1-x} = \frac{1}{2} \), the denominators are \(x\) and \(1-x\), which are not the same.
To solve this, we need to identify the common denominator. For \( \frac{1}{x} \) and \( \frac{2}{1-x} \), the common denominator is \(x(1-x)\).
This means we rewrite each fraction so that they share this common denominator:

• \( \frac{1}{x} = \frac{1-x}{x(1-x)} \)
• \( \frac{2}{1-x} = \frac{2x}{x(1-x)} \)

Doing this, the equation becomes \( \frac{1-x}{x(1-x)} - \frac{2x}{x(1-x)} = \frac{1}{2} \). Now, we can proceed to combine the fractions since they share the same denominator.

cross multiplication

After converting fractions to have a common denominator, the next step can involve cross multiplication to eliminate the fractions. In our case, we have:
\( \frac{1-3x}{x(1-x)} = \frac{1}{2} \).
By cross multiplying, we multiply both sides of the equation by \(2x(1-x)\):

• \( 2(1-3x) = x(1-x) \)

This eliminates the denominators and gives:

• \(2 - 6x = x - x^2 \)

This step is crucial because it converts a fraction problem into a standard algebraic equation, making further manipulation simpler. From here, you'll combine like terms and rearrange to form a quadratic equation.

quadratic formula

The quadratic formula is a reliable method to find the solutions for a quadratic equation of the form \(ax^2 + bx + c = 0\).
For our equation, \(-x^2 + 7x - 2 = 0\), we can identify:

• \(a = -1 \)
• \(b = 7 \)
• \(c = -2 \)

Applying these to the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Substituting in our values, we get:

• \( x = \frac{-7 \pm \sqrt{49 + 8}}{-2} \)
  This simplifies to:
• \( x = \frac{-7 \pm \sqrt{57}}{-2} \)
  Further simplified, we get the solutions:
• \( x = \frac{7 + \sqrt{57}}{2} \)
• \( x = \frac{7 - \sqrt{57}}{2} \)

These values give the two possible solutions to our original equation.