To find the possible values of $a$, we can use the distance formula. Given that the distance between the points $(a, 2a)$ and $(3, 1)$ is 3 units, we set up the equation based on the distance formula:

$\sqrt{(a - 3)^2 + (2a - 1)^2} = 3$

Step 1: Square Both Sides

Square both sides of the equation to eliminate the square root:

$(a - 3)^2 + (2a - 1)^2 = 9$

Step 2: Expand Each Term

Expand $(a - 3)^2$ and $(2a - 1)^2$:

$(a - 3)^2 = a^2 - 6a + 9$ $(2a - 1)^2 = 4a^2 - 4a + 1$

Step 3: Substitute and Combine Terms

Substitute these into the equation:

$a^2 - 6a + 9 + 4a^2 - 4a + 1 = 9$

Combine like terms:

$5a^2 - 10a + 10 = 9$

Step 4: Simplify the Equation

Subtract 9 from both sides:

$5a^2 - 10a + 1 = 0$

Step 5: Solve the Quadratic Equation

Now we have a quadratic equation in the form $5a^2 - 10a + 1 = 0$. To solve for $a$, we use the quadratic formula:

$a = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Simplify inside the square root:

$a = \frac{10 \pm \sqrt{100 - 20}}{10}$ $a = \frac{10 \pm \sqrt{80}}{10}$ $a = \frac{10 \pm 4\sqrt{5}}{10}$

Simplify further:

$a = 1 \pm \frac{2\sqrt{5}}{5}$

Final Answer

The possible values of $a$ are:

$a = 1 + \frac{2\sqrt{5}}{5} \quad \text{and} \quad a = 1 - \frac{2\sqrt{5}}{5}$

Would you like a further breakdown of each step?

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Here are 5 related questions to explore further:

1. What is the distance formula and how is it derived?
2. How can we solve quadratic equations using other methods, such as factoring or completing the square?
3. What would change if the distance between the points were different, say 5 units?
4. How does changing one coordinate affect the distance between two points?
5. How can the discriminant of a quadratic equation help determine the nature of the solutions?

Tip: When solving equations with square roots, squaring both sides is often helpful, but be mindful of potential extraneous solutions that may arise.