The equation in the image is:

$x^2 - 2ax + a^2 - b^2 = 0$

Solution:

This is a quadratic equation in $x$. To solve it, we rewrite the equation and simplify.

1. Recognize the structure:

   The equation can be written as: $x^2 - 2ax + (a^2 - b^2) = 0$

2. Factorize if possible:

   Recall the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$

   Using this, the equation becomes: $x^2 - 2ax + (a - b)(a + b) = 0$

3. Solve using the quadratic formula:

   The general quadratic equation is: $Ax^2 + Bx + C = 0$ Here, $A = 1$, $B = -2a$, and $C = a^2 - b^2$.

   Using the quadratic formula: $x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

   Substituting $A = 1$, $B = -2a$, and $C = a^2 - b^2$: $x = \frac{-(-2a) \pm \sqrt{(-2a)^2 - 4(1)(a^2 - b^2)}}{2(1)}$

   Simplify: $x = \frac{2a \pm \sqrt{4a^2 - 4(a^2 - b^2)}}{2}$ $x = \frac{2a \pm \sqrt{4a^2 - 4a^2 + 4b^2}}{2}$ $x = \frac{2a \pm \sqrt{4b^2}}{2}$ $x = \frac{2a \pm 2b}{2}$

4. Simplify the two solutions:

   $x = a + b \quad \text{or} \quad x = a - b$

Final Answer:

The solutions to the quadratic equation are: $x = a + b \quad \text{and} \quad x = a - b$

Do you want more details or have any questions about this solution?

Related Questions:

1. What happens if $a = 0$ or $b = 0$ in the equation?
2. How is the quadratic formula derived?
3. Can you graph this quadratic equation for specific values of $a$ and $b$?
4. How does the discriminant ($\Delta$) determine the nature of the roots in this case?
5. What is the significance of the factorization method used in solving the equation?

Tip:

Always verify whether you can factorize a quadratic equation before jumping to the quadratic formula. It often simplifies the solution process!