To solve for $a$ in the equation

$2(1 - a)^2 = (2 - a)^2,$

follow these steps:

1. Expand Both Sides of the Equation:

   Let's expand $(1 - a)^2$ and $(2 - a)^2$.

   • For $(1 - a)^2$: $(1 - a)^2 = 1 - 2a + a^2.$
   • For $(2 - a)^2$: $(2 - a)^2 = 4 - 4a + a^2.$

2. Substitute the Expansions Back into the Equation:

   Replace $(1 - a)^2$ and $(2 - a)^2$ with their expanded forms: $2(1 - 2a + a^2) = 4 - 4a + a^2.$

3. Distribute the 2 on the Left Side:

   Multiply each term inside the parentheses by 2: $2 - 4a + 2a^2 = 4 - 4a + a^2.$

4. Simplify the Equation:

   Bring all terms to one side to set the equation to zero. To do this, subtract $4 - 4a + a^2$ from both sides: $2 - 4a + 2a^2 - 4 + 4a - a^2 = 0.$ Combine like terms: $a^2 - 2 = 0.$

5. Solve for $a$:

   Now we have a simpler equation: $a^2 = 2.$ Take the square root of both sides: $a = \pm \sqrt{2}.$

Solution

The solutions for $a$ are: $a = \sqrt{2} \quad \text{and} \quad a = -\sqrt{2}.$

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How do you determine if both solutions for $a$ satisfy the original equation?
2. What are the implications of squaring both sides when solving an equation?
3. How would this equation change if there was an additional term on either side?
4. What other methods can be used to solve quadratic equations like this one?
5. How does factoring compare with expanding in simplifying equations?

Tip: Always check both solutions by substituting them back into the original equation to ensure they are valid.