To solve the equation:

\[
(x+2)(x-2) = (x+1)^2 + 1
\]

### Step 1: Expand both sides
First, expand the left-hand side using the difference of squares formula \((a+b)(a-b) = a^2 - b^2\):

\[
(x+2)(x-2) = x^2 - 4
\]

On the right-hand side, expand \((x+1)^2\):

\[
(x+1)^2 = x^2 + 2x + 1
\]

So the equation becomes:

\[
x^2 - 4 = x^2 + 2x + 1 + 1
\]

Simplify the right-hand side:

\[
x^2 - 4 = x^2 + 2x + 2
\]

### Step 2: Subtract \(x^2\) from both sides
Now, subtract \(x^2\) from both sides of the equation:

\[
-4 = 2x + 2
\]

### Step 3: Solve for \(x\)
Subtract 2 from both sides:

\[
-6 = 2x
\]

Now, divide by 2:

\[
x = -3
\]

### Final Answer:
\[
x = -3
\]

Would you like more details or have any questions about the solution?

Here are 5 related questions you might consider:
1. How would the solution change if the constants in the equation were different?
2. What is the geometric interpretation of solving \((x+2)(x-2) = (x+1)^2 + 1\)?
3. Can you solve a similar equation, like \((x+3)(x-3) = (x+2)^2 + 1\)?
4. What happens if you factor the right-hand side differently?
5. How do the solutions of quadratic equations change when discriminants are involved?

**Tip:** When solving equations involving squares, expanding and simplifying both sides step-by-step avoids missing important details.

Learn how to solve the quadratic equation (x+2)(x-2) = (x+1)^2 + 1 with a step-by-step guide. This problem involves algebraic techniques like factoring and expanding polynomials, making it ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation