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The problem in the image states:

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

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### 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.

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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.

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

Discover how to find all possible values of 'a' in the quadratic equation x^2 - bx + 2a = 0, where one root is twice the other. Learn step-by-step how to use the sum and product of roots to solve this problem, perfect for high school algebra students.

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