If a+b+c=7 and ab +bc+ca then find the value of a^2+b^2+c^2

We are given two equations:

1. \( a + b + c = 7 \)
2. \( ab + bc + ca = 0 \)

We are tasked with finding the value of \( a^2 + b^2 + c^2 \).

### Step 1: Use the identity for \( a^2 + b^2 + c^2 \)

The expression for \( a^2 + b^2 + c^2 \) can be written in terms of the sum and products of \(a\), \(b\), and \(c\) as follows:

\[
a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)
\]

### Step 2: Substitute the known values

Now substitute the known values from the given information:
- \( a + b + c = 7 \)
- \( ab + bc + ca = 0 \)

Substitute into the equation:

\[
a^2 + b^2 + c^2 = (7)^2 - 2(0)
\]

\[
a^2 + b^2 + c^2 = 49
\]

Thus, the value of \( a^2 + b^2 + c^2 \) is **49**.

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Would you like more details on any part of the solution? Here are five related questions:

1. How can we solve for the individual values of \(a\), \(b\), and \(c\)?
2. What is the significance of the identity \( a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca) \)?
3. Can we generalize this method for higher-degree polynomials?
4. What is the geometrical interpretation of these sums in 3D space?
5. How do we approach similar problems involving quadratic identities?

**Tip**: For any algebraic identity, try expressing it in terms of known quantities—it simplifies calculations significantly.

If a + b + c = 7 and ab + bc + ca = 0, then find the value of a^2 + b^2 + c^2.

Learn how to find the value of a^2 + b^2 + c^2 using the identity (a + b + c)^2 - 2(ab + bc + ca). Given that a + b + c = 7 and ab + bc + ca = 0, this solution is suitable for students in Grades 9-12 and explores algebraic identities in quadratic expressions.

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