Math Problem Statement

Simplify \( \frac{4x + 8}{x^2 + 9x + 20} + \frac{x + 12}{x^2 + 9x + 20} \).

Solution

Let's go through the simplification of the given expression:

4x+8x2+9x+20+x+12x2+9x+20\frac{4x + 8}{x^2 + 9x + 20} + \frac{x + 12}{x^2 + 9x + 20}

Step 1: Factor the Denominator

The common denominator in both fractions is x2+9x+20x^2 + 9x + 20. Let's factor it:

x2+9x+20=(x+4)(x+5)x^2 + 9x + 20 = (x + 4)(x + 5)

So, we can rewrite the expression as:

4x+8(x+4)(x+5)+x+12(x+4)(x+5)\frac{4x + 8}{(x + 4)(x + 5)} + \frac{x + 12}{(x + 4)(x + 5)}

Step 2: Combine the Fractions

Since both fractions have the same denominator, we can combine the numerators over the common denominator:

4x+8+x+12(x+4)(x+5)\frac{4x + 8 + x + 12}{(x + 4)(x + 5)}

Step 3: Simplify the Numerator

Combine like terms in the numerator:

4x+8+x+12=5x+204x + 8 + x + 12 = 5x + 20

So we now have:

5x+20(x+4)(x+5)\frac{5x + 20}{(x + 4)(x + 5)}

Step 4: Factor the Numerator if Possible

Notice that 5x+205x + 20 has a common factor of 5:

5x+20=5(x+4)5x + 20 = 5(x + 4)

Substitute back into the fraction:

5(x+4)(x+4)(x+5)\frac{5(x + 4)}{(x + 4)(x + 5)}

Step 5: Cancel Common Terms

Now, we can cancel x+4x + 4 from the numerator and the denominator, leaving:

5x+5\frac{5}{x + 5}

Final Answer

The simplified expression is:

5x+5\frac{5}{x + 5}

Matching with the Options

This answer matches the third option:

5x+5\frac{5}{x + 5}

Would you like further explanation or have any questions?


Here are 5 related questions to deepen understanding:

  1. How would the result change if the numerator was 4x+104x + 10 instead of 4x+84x + 8?
  2. Why is it helpful to factor both the numerator and denominator before simplifying?
  3. What would be the domain restrictions for this expression?
  4. How would you combine terms if the fractions had different denominators?
  5. What common mistakes should be avoided when simplifying expressions with multiple terms?

Tip: Always look for common factors in the numerator and denominator before canceling to simplify expressions accurately.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring

Formulas

Factoring trinomials
Combining fractions with common denominators

Theorems

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Suitable Grade Level

Grades 8-10