https://file.aiquickdraw.com/mathbot/file/1726688322448iqobt9jg.jpg

The equation in the image appears to be:

\[
(x - 3)^2 - (x + 1)^2 = x^2 - 1
\]

Let's solve this equation step by step.

### Step 1: Expand both sides

We start by expanding the squares on the left-hand side.

\[
(x - 3)^2 = x^2 - 6x + 9
\]
\[
(x + 1)^2 = x^2 + 2x + 1
\]

Now substitute these into the equation:

\[
(x^2 - 6x + 9) - (x^2 + 2x + 1) = x^2 - 1
\]

### Step 2: Simplify the left-hand side

Distribute the negative sign:

\[
x^2 - 6x + 9 - x^2 - 2x - 1 = x^2 - 1
\]

Now combine like terms:

\[
-8x + 8 = x^2 - 1
\]

### Step 3: Move all terms to one side

Move all terms to one side to form a quadratic equation:

\[
0 = x^2 + 8x - 9
\]

This simplifies to:

\[
x^2 + 8x - 9 = 0
\]

### Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 + 8x - 9 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 8\)
- \(c = -9\)

Substitute these into the quadratic formula:

\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-9)}}{2(1)}
\]
\[
x = \frac{-8 \pm \sqrt{64 + 36}}{2}
\]
\[
x = \frac{-8 \pm \sqrt{100}}{2}
\]
\[
x = \frac{-8 \pm 10}{2}
\]

Thus, we have two possible solutions for \(x\):

1. \(x = \frac{-8 + 10}{2} = \frac{2}{2} = 1\)
2. \(x = \frac{-8 - 10}{2} = \frac{-18}{2} = -9\)

### Final Answer:
The solutions to the equation are \(x = 1\) and \(x = -9\).

Would you like further details or explanations? Here are some related questions for deeper understanding:

1. What are the steps involved in factoring a quadratic equation?
2. How do you verify the solutions of a quadratic equation by substitution?
3. Could this quadratic equation be factored instead of using the quadratic formula?
4. What are the general applications of solving quadratic equations in real life?
5. What are alternative methods to solve quadratic equations apart from the quadratic formula?

**Tip:** Always check the discriminant \(b^2 - 4ac\) to determine the nature of the roots (real, complex, or equal).

Solve the quadratic equation (x - 3)^2 - (x + 1)^2 = x^2 - 1 with a step-by-step approach, covering concepts like equation expansion and solving quadratic equations using the quadratic formula. Suitable for high school students in Grades 9-10.

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