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If a+b+c=10 and ab + bc +ac=31 , then th...

If a+b+c=10 and ab + bc +ac=31 , then the value of `a^(2)+b^(2)+c^(2)` is

A

28

B

100

C

62

D

38

Text Solution

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The correct Answer is:
To find the value of \( a^2 + b^2 + c^2 \) given that \( a + b + c = 10 \) and \( ab + ac + bc = 31 \), we can use the algebraic identity: \[ a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + ac + bc) \] ### Step-by-step Solution: 1. **Identify the given values:** - \( a + b + c = 10 \) - \( ab + ac + bc = 31 \) 2. **Substitute the values into the identity:** \[ a^2 + b^2 + c^2 = (10)^2 - 2(31) \] 3. **Calculate \( (10)^2 \):** \[ (10)^2 = 100 \] 4. **Calculate \( 2(31) \):** \[ 2(31) = 62 \] 5. **Subtract \( 62 \) from \( 100 \):** \[ a^2 + b^2 + c^2 = 100 - 62 = 38 \] 6. **Final answer:** \[ a^2 + b^2 + c^2 = 38 \]
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