Let's solve the equation \(\left(x+3\right)^2 = 64\) step by step.

### Step 1: Take the square root of both sides
To eliminate the square on the left-hand side, take the square root of both sides of the equation:

\[
\sqrt{\left(x+3\right)^2} = \sqrt{64}
\]

This simplifies to:

\[
x + 3 = \pm 8
\]

### Step 2: Solve for \(x\)
Now, solve the equation for \(x\) by considering both the positive and negative square roots:

1. \(x + 3 = 8\)
   \[
   x = 8 - 3 = 5
   \]

2. \(x + 3 = -8\)
   \[
   x = -8 - 3 = -11
   \]

### Final Answer:
The solutions to the equation are:

\[
x = 5 \quad \text{or} \quad x = -11
\]

Would you like more details on any part of the solution, or do you have any questions?

### Follow-Up Questions:
1. How do you verify the solutions of a quadratic equation like this one?
2. What happens if the equation was \(\left(x+3\right)^2 = 0\)?
3. How do you graph the solutions on a number line?
4. What is the significance of taking both the positive and negative square roots?
5. How would the equation change if it were \(\left(x+3\right)^2 = -64\)?

### Tip:
Always consider both the positive and negative roots when solving equations of the form \(a^2 = b\), as they can lead to two different solutions.

Learn how to solve the equation (x + 3)^2 = 64 step-by-step with detailed explanations and solutions. This problem covers key algebraic concepts, including the quadratic equation formula and the square root theorem. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation