Solving for \((a + b + c)\) with Given Algebraic Conditions
The problem asks us to find the value of \((a + b + c)\) given two conditions involving variables \(a\), \(b\), and \(c\). The conditions are:
- Condition 1: \(\left(a-b\right)^2 + \left(b - c\right)^2 + \left(c-a\right)^2 = 6\)
- Condition 2: \(a^2 + b^2+c^2 = 29\)
Expanding the First Condition
Let's start by expanding the terms in the first condition using the algebraic identity \((x-y)^2 = x^2 - 2xy + y^2\):
- \(\left(a-b\right)^2 = a^2 - 2ab + b^2\)
- \(\left(b-c\right)^2 = b^2 - 2bc + c^2\)
- \(\left(c-a\right)^2 = c^2 - 2ca + a^2\)
Now, sum these expanded terms:
\(\left(a^2 - 2ab + b^2\right) + \left(b^2 - 2bc + c^2\right) + \left(c^2 - 2ca + a^2\right) = 6\)
Combine like terms:
\(2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 6\)
Factor out the common factor of 2:
\(2\left(a^2 + b^2 + c^2\right) - 2\left(ab + bc + ca\right) = 6\)
Using the Second Condition
We are given that \(a^2 + b^2 + c^2 = 29\). Substitute this value into the equation derived above:
\(2(29) - 2\left(ab + bc + ca\right) = 6\)
\(58 - 2\left(ab + bc + ca\right) = 6\)
Now, solve for the term \(\left(ab + bc + ca\right)\):
\(58 - 6 = 2\left(ab + bc + ca\right)\)
\(52 = 2\left(ab + bc + ca\right)\)
Divide both sides by 2:
\(ab + bc + ca = \frac{52}{2}\)
\(ab + bc + ca = 26\)
Finding \((a + b + c)\)
We know the algebraic identity for the square of a sum:
\(\left(a + b + c\right)^2 = a^2 + b^2 + c^2 + 2\left(ab + bc + ca\right)\)
Substitute the known values of \(a^2 + b^2 + c^2 = 29\) and \(ab + bc + ca = 26\) into this identity:
\(\left(a + b + c\right)^2 = 29 + 2(26)\)
\(\left(a + b + c\right)^2 = 29 + 52\)
\(\left(a + b + c\right)^2 = 81\)
To find \(\left(a + b + c\right)\), take the square root of both sides:
\(a + b + c = \pm \sqrt{81}\)
\(a + b + c = \pm 9\)
Conclusion
Based on the given conditions and algebraic manipulations, the value of \((a + b + c)\) is \(\pm 9\).
Similar Questions
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Important Questions from Algebric Equations
A group of 630 children is seated in rows for a group photo session. Each row contains three less children than the row in front of it. Which one of the following number of rows is not possible?
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What is the integer assigned to N?
Four persons, Alok, Bhupesh, Chander and Dinesh have a total of Rs. 100 among themselves. Alok and Bhupesh between them have as much money as Chander and Dinesh between them, but Alok has more money than Bhupesh; and Chander has only half the money that Dinesh has. Alok has in fact Rs. 5 more than Dinesh has.
Who has the maximum amount of money?
If x=3/2, then the value of 27x3-54x2+36x-11 is
If a+b+c = 6 and ab+bc+ca = 11, then the value of bc(b+c) + ca(c+a) +ab(a+b) +3abc is
