Math Problem Statement
if a+b+c = 0 and a^2 + b^2 + c^2 = sqrt(74) what is a^4 + b^4 + c^4? Explain your working step by step
Solution
We are tasked to find given the conditions:
Step 1: Expand and simplify
The sum can be written using the following relation:
Thus, we need:
- (already given as ).
- .
Step 2: Use the condition
From , the square of the sum is: Simplify: Substitute : Solve for :
Step 3: Calculate
We use the identity: Since , the term vanishes. Thus: Substitute :
Step 4: Compute
Using the formula from Step 1: Substitute and : Simplify:
Final Answer:
Questions to Reflect On:
- Why does simplify the problem significantly?
- How is the symmetric sum derived using basic algebraic identities?
- What role does the condition play in this calculation?
- Can this method be generalized for other values of and ?
- What would the result be if ?
Tip:
Always start by simplifying given conditions, as they often reduce the complexity of the problem through relationships among the variables.
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Math Problem Analysis
Mathematical Concepts
Algebra
Symmetric Functions
Polynomials
Formulas
a^4 + b^4 + c^4 = (a^2 + b^2 + c^2)^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
a^2b^2 + b^2c^2 + c^2a^2 = (ab + bc + ca)^2 - 2abc(a + b + c)
Theorems
Expansion of polynomial identities
Symmetric sum properties
Suitable Grade Level
Grades 10-12
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