The problem in the image states:

One root of $x^2 - bx + 2a = 0$ is twice the other. Find all possible values of $a$.

Step-by-Step Solution:

1. Let the Roots Be Represented Mathematically: Let the roots of the quadratic equation $x^2 - bx + 2a = 0$ be $r$ and $2r$, where one root is twice the other.

2. Sum of Roots: Using the sum of roots formula: r + 2r = b \quad \text{(since the sum of roots is equal to the coefficient of x with a negative sign)}. Thus: $3r = b \quad \text{or equivalently} \quad r = \frac{b}{3}.$

3. Product of Roots: Using the product of roots formula: $r \cdot 2r = 2a \quad \text{(since the product of the roots is equal to the constant term)}.$ Thus: $2r^2 = 2a \quad \text{or equivalently} \quad r^2 = a.$

4. Substitute $r = \frac{b}{3}$ into $r^2 = a$: Since $r = \frac{b}{3}$, substitute it into $r^2 = a$: $\left(\frac{b}{3}\right)^2 = a.$ Simplify: $\frac{b^2}{9} = a.$

5. Express $a$ in Terms of $b$: $a = \frac{b^2}{9}.$

Final Answer:

All possible values of $a$ are given by $a = \frac{b^2}{9}$, where $b$ is any real number.

Would you like a further explanation or additional examples? Here are 5 related questions for practice:

1. How do you find the roots of a quadratic equation given its coefficients?
2. What are the conditions for two roots of a quadratic equation to be related linearly (e.g., one being a multiple of the other)?
3. How do the sum and product of roots relate to the coefficients of a quadratic equation?
4. What happens to the roots of a quadratic equation if you double the constant term $c$?
5. Solve the quadratic equation $x^2 - 6x + 8 = 0$ and verify the relationships of sum and product of roots.

Tip: Always verify the relationship between roots and coefficients when solving quadratic equations.