Question

In Exercises 33–68, add or subtract as indicated. 4/(x2+6x+9) + 4/(x+3)

Accepted Answer

Step 1: Recognize that the denominators in the given expression are different. The first term has a denominator of $$x^2 + 6x + 9$$, and the second term has a denominator of $$x + 3. $$Factorize $$x^2 + 6x + 9 to $$simplify the expression. Notice that $$x^2 + 6x + 9 is a $$perfect square trinomial, so it can be rewritten as $$(x + 3)^2. $$Step 2: Rewrite the expression using the factored form of the first denominator: $$\frac{4}{(x + 3)^2} + \frac{4}{x + 3}. $$Now, observe that the denominators are $$(x + 3)^2$$ and $$x + 3$$, which are not the same. To combine these fractions, we need a common denominator. Step 3: Determine the least common denominator (LCD). The LCD for $$(x + 3)^2$$ and $$x + 3 is$$ $$(x + 3)^2. $$Rewrite the second fraction $$\frac{4}{x + 3} so $$that it has the LCD as its denominator. Multiply both the numerator and denominator of $$\frac{4}{x + 3} by$$ $$x + 3$$, resulting in $$\frac{4(x + 3)}{(x + 3)^2}. $$Step 4: Combine the two fractions under the common denominator $$(x + 3)^2. $$The first fraction remains $$\frac{4}{(x + 3)^2}$$, and the second fraction becomes $$\frac{4(x + 3)}{(x + 3)^2}. $$Add the numerators: $$4 + 4(x + 3)$$, keeping the denominator $$(x + 3)^2. $$Step 5: Simplify the numerator. Distribute the $$4 in$$ $$4(x + 3) to $$get $$4x + 12. $$Combine this with the $$4$$ from the first term to get $$4x + 16. $$The final expression is $$\frac{4x + 16}{(x + 3)^2}.$$

In Exercises 33–68, add or subtract as indicated. 4/(x²+6x+9) + 4/(x+3)

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Concetti chiave

Factoring Quadratics

Finding a Common Denominator

Simplifying Rational Expressions