Problem 33 - Intermediate Algebra

Question

Solve equation. \(\frac{1}{3 x-18}+\frac{5}{6-x}=\frac{1}{3}\)

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Identify a Common Denominator

The denominators in the given equation are \(3x - 18\) and \(6 - x\). We must find a common denominator to combine the fractions. The expression \((3x - 18)(6 - x)\) serves as the common denominator.

Step 2: Multiply Equation by Common Denominator

Multiply the entire equation by the common denominator \((3x - 18)(6 - x)\) to eliminate the denominators:\[(3x - 18)(6 - x) \left( \frac{1}{3x - 18} + \frac{5}{6 - x} \right) = \frac{1}{3} (3x - 18)(6 - x)\]This simplifies to:\[(6 - x) + 5(3x - 18) = \frac{1}{3}(3x - 18)(6 - x)\]

Step 3: Distribute and Simplify

Distribute the terms on both sides:\[(6 - x) + 15x - 90 = \frac{1}{3}(18x - 3x^2 - 108 + 6x)\]Combine like terms:\[15x - x - 90 + 6 = \frac{1}{3}(-3x^2 + 24x - 108)\]Simplify further:\[14x - 84 = -x^2 + 8x - 36\]

Step 4: Move All Terms to One Side

Rearrange all terms to form a quadratic equation:\[0 = x^2 - 6x + 48\]Which can be rewritten as:\[ x^2 - 6x + 48 = 0 \]

Step 5: Use the Quadratic Formula

Solve the quadratic equation using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \(a = 1\), \(b = -6\), and \(c = 48\). Substitute these values:\[x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 48}}{2 \cdot 1} \]This simplifies to:\[x = \frac{6 \pm \sqrt{36 - 192}}{2} \]Since the expression under the square root is negative (\(36 - 192 = -156\)), the roots are complex and not real.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula is given by:

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

This formula directly provides the solutions (or roots) of the quadratic equation. Here, \( a \), \( b \), and \( c \) represent the coefficients of the terms in the quadratic equation. The symbol \( \pm \) indicates there are generally two solutions: one involving addition and the other subtraction.

To use the quadratic formula effectively, identify the coefficients \( a \), \( b \), and \( c \) from the equation. Substitute these values into the formula to solve for \( x \). The expression inside the square root, called the discriminant \( b^2 - 4ac \), determines the nature of the roots. Depending on this value, the roots could be real or complex numbers.

Complex Roots

When solving quadratic equations using the quadratic formula, the discriminant \( b^2 - 4ac \) plays a critical role. If the discriminant is negative, as in our exercise where \( 36 - 192 = -156 \), the roots of the quadratic equation are complex. Complex numbers include both a real part and an imaginary part.

Complex roots occur in conjugate pairs. This means if one solution is \( p + qi \), the other is \( p - qi \), where \( p \) and \( q \) are real numbers and \( i \) is the imaginary unit satisfying \( i^2 = -1 \). Understanding complex roots is important because sometimes equations do not have a solution within the set of real numbers. Recognizing when to expect such results is key to mastering quadratic equations.

Common Denominator

When dealing with rational equations, one of the first steps is to find a common denominator. In the given exercise, the denominators were \( 3x - 18 \) and \( 6 - x \). A common denominator is a shared multiple of these individual denominators, which allows you to combine the fractions into a single expression.

The process of finding a common denominator involves:

• Identifying each unique factor in the denominators.
• Creating a new denominator by multiplying these factors together.

In this case, the expression \((3x - 18)(6 - x)\) served as the common denominator. This step is crucial as it allows you to eliminate the fractions by multiplying each term by this common denominator, simplifying the equation into a more manageable form without fractions.

Rational Equation Simplification

Simplifying rational equations involves a series of methodical steps. Initially, after identifying the common denominator, as demonstrated in the exercise, you should multiply the entire equation by this common denominator to eliminate the fractional components.

Once the fractions are eliminated, the next step is to distribute the multiplication over addition or subtraction and then combine like terms. This can lead to a simplified version of the original equation. Often, this results in a form that resembles a standard polynomial equation which is simpler to solve. In the given exercise, the simplification process led to a quadratic equation, \( x^2 - 6x + 48 = 0 \).

By transforming a rational equation into a polynomial equation, you make it more straightforward to solve using established methods like factoring, completing the square, or employing the quadratic formula. The ultimate goal of this process is to find the values of the variable that satisfy the original equation.