Problem 30 - Algebra 2

Question

Simplify each expression. $$ \frac{5}{x^{2}-3 x-28}+\frac{7}{2 x-14} $$

Step-by-Step Solution

Answer

\( \frac{7x + 38}{2(x-7)(x+4)} \)

Step 1: Factor the Denominators

Start by factoring each denominator separately. For the first expression, the denominator is a quadratic expression: \[ x^2 - 3x - 28 = (x - 7)(x + 4) \] For the second expression, identify the common factor:\[ 2x - 14 = 2(x - 7) \] Now we have: \[ \frac{5}{(x-7)(x+4)} + \frac{7}{2(x-7)} \] .

Step 2: Determine the Least Common Denominator (LCD)

The common denominator must cover all different factors of the denominators. In this case, it's:\[ (x-7)(x+4) \times 2 \]Thus, the LCD is \[ 2(x-7)(x+4) \].

Step 3: Adjust the Numerators

Express each fraction with the LCD as the new denominator. Multiply the numerators by the missing factors:For the first fraction:\[ \frac{5}{(x-7)(x+4)} \times \frac{2}{2} = \frac{10}{2(x-7)(x+4)} \]For the second fraction:\[ \frac{7}{2(x-7)} \times \frac{(x+4)}{(x+4)} = \frac{7(x+4)}{2(x-7)(x+4)} \] .

Step 4: Add the Fractions

Combine the two fractions using their common denominator:\[ \frac{10}{2(x-7)(x+4)} + \frac{7(x+4)}{2(x-7)(x+4)} \]Since the denominators are the same, you can add the numerators:\[ \frac{10 + 7(x + 4)}{2(x-7)(x+4)} \] .

Step 5: Simplify the Numerator

Simplify the combined numerator expression:\[ 10 + 7(x + 4) = 10 + 7x + 28 \]Combine like terms:\[ 7x + 38 \] .

Step 6: Final Expression

The expression becomes:\[ \frac{7x + 38}{2(x-7)(x+4)} \]This is the simplified form of the original expression.

Key Concepts

Factoring Quadratics

Rational expressions often involve quadratic expressions, and learning to factor them is crucial in simplifying such expressions. Factoring a quadratic means rewriting it as a product of two binomials. It involves finding two numbers that multiply to give the constant term (the last number in the expression) and add to give the middle term coefficient (the coefficient of the linear term). For example, with the quadratic \(x^2 - 3x - 28\), the two numbers sought are -7 and 4. Why? Because \(-7 \cdot 4 = -28\) and \(-7 + 4 = -3\).
When you see \(x^2 - 3x - 28\), factor it as \((x - 7)(x + 4)\). If you're factoring a simple linear expression, like \(2x - 14\), you can take out the greatest common factor, here it results in \(2(x - 7)\). This foundational step makes the following steps possible by enabling a common ground to combine the fractions.

Least Common Denominator

To add or subtract rational expressions, you must have a common denominator, much like regular fractions. Here, the idea of a "Least Common Denominator" (LCD) comes in handy. It's essentially the smallest expression that includes all denominators involved, covering all unique factors at their highest occurrence.
For our problem, we had two denominators: \((x-7)(x+4)\) and \(2(x-7)\). The LCD should include all factors, leading us to \(2(x-7)(x+4)\). The factor \(2\) covers the second expression's unique coefficient, while the variables \((x-7)\) and \((x+4)\) are necessary to ensure full coverage of both original denominators. This serves as the base for adjusting all numerators to match, allowing seamless combination.

Combining Fractions

Once the least common denominator is determined, the next step is to express each fraction with this common denominator. This involves adjusting each numerator to reflect the multiplication of its associated denominator into the LCD. For each fraction:

• Multiply numerators by what their denominator lacks to reach the LCD.
• Maintain the equivalent value by multiplying both the numerator and the denominator by the same factor.

In our example:

• The fraction \(\frac{5}{(x-7)(x+4)}\) was multiplied by \(\frac{2}{2}\), becoming \(\frac{10}{2(x-7)(x+4)}\).
• The other fraction \(\frac{7}{2(x-7)}\) was multiplied by \(\frac{(x+4)}{(x+4)}\), resulting in \(\frac{7(x+4)}{2(x-7)(x+4)}\).

Both expressions now share the same denominator, making addition straightforward.

Simplifying Algebraic Expressions

Finally, once the fractions are expressed with a common denominator, adding them becomes straightforward. The main task here is to combine the numerators into a single fraction. With both denominators identical, you can add the numerators directly. For the example, combining gives:

• \(10 + 7(x + 4)\), which simplifies to \(10 + 7x + 28\).

Combine like terms carefully, ensuring all constants and coefficients are addressed, leading to a simplified expression: \(7x + 38\).
Ultimately, the expression simplifies to: \(\frac{7x + 38}{2(x-7)(x+4)}\). This highlights the entire process, reinforcing each concept involved in practical application. Simplifying ensures the expression is in its simplest form, which is essential in both practical mathematics and when checking work for errors or alternative approaches.