Solve (a+b)(a-b) = (a+b)²: Finding Values of a and b

Find $a ,b$ such that:

$(a+b)(a-b)=(a+b)^2$

To solve this problem, we'll follow these steps:

• Step 1: Expand both sides of the given equation.
• Step 2: Compare and simplify the resulting expressions.
• Step 3: Solve for $a$ and $b$.

Now, let's work through each step:
Step 1: The given equation is $(a+b)(a-b) = (a+b)^2$. Let's expand both sides:
- Left side: $(a+b)(a-b) = a^2 - b^2$ based on the difference of squares formula.
- Right side: $(a+b)^2 = a^2 + 2ab + b^2$ using the square of a sum formula.

Step 2: Setting the expanded forms equal gives us:
$a^2 - b^2 = a^2 + 2ab + b^2$.

Step 3: Simplify and solve the equation:
- Subtract $a^2$ from both sides: $-b^2 = 2ab + b^2$.
- Add $b^2$ to both sides: $0 = 2ab + 2b^2$.
- Factor the right-hand side: $0 = 2b(a + b)$.

This gives us two possible conditions:
1) $2b = 0$, which implies $b = 0$.
2) $a + b = 0$, which implies $a = -b$.

Since $a = -b$ satisfies the equation for any $a$ if $b$ is not zero, and when $b = 0$, the equation simplifies to $0 = 0$, both conditions are valid.

Therefore, the solutions are $a = -b$ or $b = 0$.

In conclusion, the answer is: $a=-b$ or $0=b$.