We are given the equation:

$a^2 + (a + b)^2 + (a + c)^2 = 410$

We need to find the value of $a + b + c$.

Step 1: Expand the squared terms

Let's expand the squared terms:

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

$(a + c)^2 = a^2 + 2ac + c^2$

Substitute these into the original equation:

$a^2 + \left(a^2 + 2ab + b^2\right) + \left(a^2 + 2ac + c^2\right) = 410$

Step 2: Simplify the equation

Combine like terms:

$a^2 + a^2 + 2ab + b^2 + a^2 + 2ac + c^2 = 410$

This simplifies to:

$3a^2 + 2ab + 2ac + b^2 + c^2 = 410$

Step 3: Assume a specific value for $a$

Let's assume $a = 10$ and check if it works:

Substitute $a = 10$ into the equation:

$3(10)^2 + 2(10)b + 2(10)c + b^2 + c^2 = 410$

$3(100) + 20b + 20c + b^2 + c^2 = 410$

$300 + 20b + 20c + b^2 + c^2 = 410$

Step 4: Simplify further

Subtract 300 from both sides:

$20b + 20c + b^2 + c^2 = 110$

Step 5: Solve the simplified equation

Let’s assume $b + c = x$ and $bc = y$. Then:

$20x + x^2 = 110$

We now solve the quadratic equation:

$x^2 + 20x - 110 = 0$

The roots of this equation will give us the value of $x = b + c$.

Using the quadratic formula:

$x = \frac{-20 \pm \sqrt{20^2 + 4 \times 110}}{2}$

$x = \frac{-20 \pm \sqrt{400 + 440}}{2}$

$x = \frac{-20 \pm \sqrt{840}}{2}$

$x = \frac{-20 \pm 2\sqrt{210}}{2}$

$x = -10 \pm \sqrt{210}$

This implies that $b + c = \sqrt{210} - 10$ (we discard the negative root as it leads to negative values for $b$ or $c$).

Step 6: Find $a + b + c$

Finally, we calculate:

$a + b + c = 10 + (\sqrt{210} - 10) = \sqrt{210}$

So, $a + b + c = \sqrt{210}$.

Conclusion:

Thus, the value of $a + b + c$ is $\sqrt{210}$.

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Would you like more details or have any other questions?

Here are some related questions:

1. What happens if you choose a different value for $a$?
2. How would the problem change if there was a constant added to the equation?
3. Can we factor the quadratic equation differently?
4. How would you check if the value of $b$ and $c$ are correct?
5. What other methods can be used to solve this type of problem?

Tip: When dealing with quadratic equations, always check both possible roots and evaluate their practicality in the context of the problem.