https://file.aiquickdraw.com/mathbot/file/1726643978343w5elhahs.jpg

The equation shown in the image is:

\[
(x - 3)^2 - x^2 = x - 19
\]

Let's solve this step by step.

### Step 1: Expand \((x - 3)^2\)
\[
(x - 3)^2 = x^2 - 6x + 9
\]

So, the equation becomes:
\[
(x^2 - 6x + 9) - x^2 = x - 19
\]

### Step 2: Simplify the equation
The \(x^2\) terms cancel out:
\[
-6x + 9 = x - 19
\]

### Step 3: Rearrange the equation
Move all terms to one side:
\[
-6x - x = -19 - 9
\]
\[
-7x = -28
\]

### Step 4: Solve for \(x\)
\[
x = \frac{-28}{-7} = 4
\]

### Solution:
The value of \(x\) is \(4\).

Would you like more details on any step? Here are five related questions to deepen your understanding:

1. How do you expand binomials like \((x - 3)^2\)?
2. What happens when terms like \(x^2\) cancel out in an equation?
3. How do you solve linear equations after simplifying?
4. What methods can be used to check the solution of this type of equation?
5. How can we verify the final solution by substituting it back into the original equation?

**Tip**: Always verify your solution by plugging it back into the original equation to ensure it's correct.

Learn how to solve the algebraic equation (x - 3)^2 - x^2 = x - 19 with a detailed step-by-step solution. This problem covers key algebraic concepts, including binomial expansion and linear equation solving. Suitable for students in Grades 9-11.

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