a^2 + (a + b)^2 + (a + c)^2 = 410, find a + b + c.

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 \( \sqrt{210} \).

---

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.

Learn how to solve the equation a^2 + (a + b)^2 + (a + c)^2 = 410 with a detailed step-by-step solution. This problem involves expanding quadratic terms and applying algebraic manipulation to find the values of a, b, and c. Suitable for advanced high school students.

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