Problem 45 - Algebra 2

Question

Simplify each expression. $$ \frac{x}{x^{2}+5 x+6}-\frac{2}{x^{2}+4 x+4} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x^2 - 6}{(x + 2)^2(x + 3)}\).

Step 1: Identify and Factor Denominators

First, look at each denominator in the expression and determine if they can be factored. For the first fraction, the denominator is \(x^2 + 5x + 6\). Factoring this quadratic, we find \((x + 2)(x + 3)\). For the second fraction, the denominator is \(x^2 + 4x + 4\), which factors to \((x + 2)^2\).

Step 2: Find a Common Denominator

The factored form of both denominators gives us: - First denominator: \((x + 2)(x + 3)\)- Second denominator: \((x + 2)^2\)The least common denominator (LCD) is \((x + 2)^2(x + 3)\) since it covers all the factors needed for both denominators.

Step 3: Rewrite Each Fraction with the LCD

Rewrite each term so that their denominators match the LCD. - For \(\frac{x}{(x + 2)(x + 3)}\), multiply the numerator and denominator by \((x + 2)\) to get \(\frac{x(x + 2)}{(x + 2)^2(x + 3)}\).- For \(\frac{2}{(x + 2)^2}\), multiply the numerator and denominator by \((x + 3)\) to get \(\frac{2(x + 3)}{(x + 2)^2(x + 3)}\).

Step 4: Subtract the Fractions

Now that both fractions have the same denominator, we can subtract them:\[\frac{x(x + 2)}{(x + 2)^2(x + 3)} - \frac{2(x + 3)}{(x + 2)^2(x + 3)}\]Subtract the numerators:\[x(x+2) - 2(x+3) = x^2 + 2x - 2x - 6 = x^2 - 6\]

Step 5: Simplify the Expression

Write the result of the subtraction:\[\frac{x^2 - 6}{(x + 2)^2(x + 3)}\]Check if the numerator can be further factored or cancelled with the denominator. In this case, \(x^2 - 6\) cannot be factored with existing factors in the denominator, so the expression is fully simplified.

Key Concepts

Factoring Quadratics

Factoring quadratics is a process of breaking down a quadratic equation into simpler terms using multiplication of binomials. Quadratics generally have the form \(ax^2 + bx + c\). To factor these:

• Identify two numbers that multiply to give \(c\) and add to give \(b\).
• Rewrite the middle term using these two numbers and factor by grouping.

In the exercise, for the first expression, \(x^2 + 5x + 6\) was factored as \((x + 2)(x + 3)\). Here, 2 and 3 multiply to 6 and add to 5. For the second expression, \(x^2 + 4x + 4\) factors into \((x + 2)^2\), as both factors are the same.

Least Common Denominator

Finding the least common denominator (LCD) is essential when working with fractions that have different denominators. The LCD is the smallest expression that each original denominator can divide into evenly.

• Factor each denominator completely.
• Take each factor to the highest power it occurs in any one of the denominators.

In this exercise, the denominators were \((x + 2)(x + 3)\) and \((x + 2)^2\). The LCD, therefore, is \((x + 2)^2(x + 3)\), since it includes both \((x + 2)\) squared from the second denominator and \((x + 3)\) from the first.

Subtracting Fractions

To subtract fractions, start by ensuring both have the same denominator. Once that is done, subtract the numerators directly and keep the common denominator.

• Rewrite each fraction to have the common denominator, which was done by multiplying the numerators by the necessary factor.
• Once rewritten, subtract the newly constructed numerators.

In the worked example, \(\frac{x(x + 2)}{(x + 2)^2(x + 3)} - \frac{2(x + 3)}{(x + 2)^2(x + 3)}\) led to a subtraction of numerators: \(x(x+2) - 2(x+3) = x^2 - 6\).

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. This often includes:

• Combining like terms.
• Checking for any common factors that can be canceled out between the numerator and denominator.

In our example, the expression \(\frac{x^2 - 6}{(x + 2)^2(x + 3)}\) cannot be simplified further, as the numerator \(x^2 - 6\) does not have common factors with the denominator. It remains as it is since further factoring wasn't possible.