Problem 20 - Algebra and Trigonometry

Question

\(5-60\) Find all real solutions of the equation. $$ \frac{10}{x}-\frac{12}{x-3}+4=0 $$

Step-by-Step Solution

Answer

The real solutions are \( x = 5.25 \) and \( x = -1.75 \).

Step 1: Find a Common Denominator

To combine the fractions \( \frac{10}{x} \) and \( \frac{12}{x-3} \), we need to find a common denominator. The denominators are \( x \) and \( x-3 \), so the least common denominator is \( x(x-3) \).

Step 2: Write Each Term with the Common Denominator

Rewrite the fractions with the common denominator: \( \frac{10(x-3)}{x(x-3)} \) and \( \frac{12x}{x(x-3)} \). The equation becomes: \[ \frac{10(x-3)}{x(x-3)} - \frac{12x}{x(x-3)} + \frac{4x(x-3)}{x(x-3)} = 0 \]

Step 3: Combine the Fractions

Combine the numerators under the common denominator: \[ \frac{10(x-3) - 12x + 4x(x-3)}{x(x-3)} = 0 \] The common denominator allows us to combine the fractions.

Step 4: Simplify the Numerator

Simplify the expression in the numerator: First expand: \( 10(x-3) = 10x - 30 \), \( 12x \), \( 4x(x-3) = 4x^2 - 12x \).Combine these: \[ 10x - 30 - 12x + 4x^2 - 12x = 4x^2 - 14x - 30 \]

Step 5: Solve the Quadratic Equation

We now have: \[ \frac{4x^2 - 14x - 30}{x(x-3)} = 0 \] Set the numerator equal to zero: \[ 4x^2 - 14x - 30 = 0 \]

Step 6: Find Roots Using the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \). Here, \( a = 4 \), \( b = -14 \), and \( c = -30 \).Calculate: \[ x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(4)(-30)}}{2(4)} \] This simplifies to: \[ x = \frac{14 \pm \sqrt{196 + 480}}{8} \] \[ x = \frac{14 \pm 28}{8} \]

Step 7: Calculate Possible Solutions

The potential solutions are: \[ x = \frac{42}{8} = 5.25 \] and \[ x = \frac{-14}{8} = -1.75 \].

Step 8: Check for Domain Restrictions

Check if the solutions lie within the domain of the original equation. The values \( x = 5.25 \) and \( x = -1.75 \) don't make any denominator zero (i.e., \( x eq 0, x eq 3 \)), so both are real solutions.

Key Concepts

Common Denominator

In mathematics, particularly when working with fractions, it's pivotal to have a shared baseline for comparison or combination. This is the common denominator. When given an equation such as \( \frac{10}{x} - \frac{12}{x-3} + 4 = 0 \), finding a common denominator simplifies the manipulation of the fractions involved.
A common denominator is essentially the least common multiple (LCM) of all denominators in the problem. Here, we have denominators \( x \) and \( x-3 \).
To find a common denominator, multiply these together: \( x(x-3) \).
This step ensures each fraction can be rewritten over a unified base, allowing for straightforward combination and simplification of terms later.

Quadratic Formula

The quadratic formula is a vital tool for finding the roots of a quadratic equation, which is any equation in the form \( ax^2 + bx + c = 0 \).
The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]It allows us to calculate the values of \( x \) that satisfy the equation, known as the roots.
In the given exercise, after simplifying, the quadratic equation is \( 4x^2 - 14x - 30 = 0 \).
By setting \( a = 4 \), \( b = -14 \), and \( c = -30 \), the quadratic formula becomes a straightforward substitution problem. The discriminant, \( b^2 - 4ac \), under the square root determines the nature and number of solutions:

• If positive, two real solutions exist.
• If zero, one real solution exists.
• If negative, no real solutions exist.

This formula simplifies what could otherwise be complex factoring or trial-and-error processes.

Combining Fractions

Once a common denominator is established, fractions can be combined easily by addressing only their numerators.
Instead of handling separate fractions, we will work with a single expression by unifying them under one denominator. In our example, once rewritten, it appears as:\[ \frac{10(x-3) - 12x + 4x(x-3)}{x(x-3)} = 0 \]This simplifies the problem to operating on the numerators:

• Expand each term: \( 10(x-3) = 10x - 30 \)
• Combine terms: \( 10x - 30 - 12x + 4x^2 - 12x \)
• Simplify: \( 4x^2 - 14x - 30 \)

With the numerators combined and simplified, solving the equation becomes a matter of focusing only on the numerator.

Domain Restrictions

Domain restrictions are crucial in equations containing fractions, to avoid undefined expressions, particularly division by zero.
At the outset, identify terms in the denominator that could lead to zero values. In this problem, denominators \( x \) and \( x-3 \) dictate the restrictions:

• \( x eq 0 \)
• \( x eq 3 \)

These restrictions help us confirm the validity of possible solutions.
After finding potential solutions from solving \( 4x^2 - 14x - 30 = 0 \), check these against the domain restrictions to make sure they do not invalidate the original setup by equaling zero in any denominator term. Valid solutions from the example, \( x = 5.25 \) and \( x = -1.75 \), satisfy these constraints, confirming them as real and applicable solutions.