If a+1/(a+2)=3 then solve (a+2)^5-1/(a+2)^5

We are given the equation:

\[
a + \frac{1}{a + 2} = 3
\]

### Step 1: Solve for \( a \)

First, subtract \( a \) from both sides of the equation:

\[
\frac{1}{a + 2} = 3 - a
\]

Next, take the reciprocal of both sides to isolate \( a + 2 \):

\[
a + 2 = \frac{1}{3 - a}
\]

Multiply both sides by \( (3 - a) \) to eliminate the denominator:

\[
(a + 2)(3 - a) = 1
\]

Now, expand the left-hand side:

\[
3a + 6 - a^2 - 2a = 1
\]

Simplify:

\[
a + 6 - a^2 = 1
\]

Rearrange the equation:

\[
-a^2 + a + 6 = 1
\]

Move all terms to one side:

\[
-a^2 + a + 5 = 0
\]

Multiply by \(-1\) to simplify:

\[
a^2 - a - 5 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
a = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-5)}}{2(1)}
\]

Simplify:

\[
a = \frac{1 \pm \sqrt{1 + 20}}{2}
\]

\[
a = \frac{1 \pm \sqrt{21}}{2}
\]

Thus, the two possible values for \( a \) are:

\[
a = \frac{1 + \sqrt{21}}{2} \quad \text{or} \quad a = \frac{1 - \sqrt{21}}{2}
\]

### Step 2: Solve for \( (a + 2)^5 - \frac{1}{(a + 2)^5} \)

Recall from earlier that:

\[
a + 2 = \frac{1}{3 - a}
\]

Now, raise both sides of this equation to the 5th power:

\[
(a + 2)^5 = \left( \frac{1}{3 - a} \right)^5
\]

So, we need to find:

\[
(a + 2)^5 - \frac{1}{(a + 2)^5}
\]

Since \( (a + 2)^5 = \left( \frac{1}{3 - a} \right)^5 \), this expression simplifies to:

\[
\boxed{0}
\]

### Final Answer:

\[
(a + 2)^5 - \frac{1}{(a + 2)^5} = 0
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on the concept:
1. How do you solve quadratic equations using factorization?
2. What is the significance of the quadratic formula?
3. Can you solve a cubic equation using a similar approach?
4. How do reciprocal functions behave in different equations?
5. How would the solution change if \( a + \frac{1}{a+2} = 4 \)?

**Tip:** Always check if equations can be simplified before solving to reduce complexity!

Explore how to solve the equation a + 1/(a + 2) = 3 and find the value of (a + 2)^5 - 1/(a + 2)^5 with a detailed step-by-step explanation. This problem covers key algebraic concepts such as quadratic equations, reciprocal functions, and exponentiation, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation