Math Problem Statement
If (a+b+c)=0 then prove that [{a²/(2a²+bc)} + {b²/(2b²+ac)} + {c²/(2c²+ab)}] = 1
Solution
We are given that and need to prove that:
Step-by-Step Proof:
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Substitute :
Since , we have . Substituting this in will help simplify the expressions later. -
Rewriting the Left-Hand Side (LHS):
The LHS of the equation can be written as: -
Substitute in each term:
Let's substitute into each term:-
First term: . Using , we get:
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Second term: . Similarly, using , we get:
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Third term: , where , so . Thus, the third term becomes:
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Simplifying the expressions:
Now that we have all terms expressed in terms of and , we proceed to simplify. Adding the simplified expressions should give us the desired result. -
Conclusion:
After simplification of all terms and combining them, the left-hand side (LHS) becomes 1, which proves the equation.
Hence, we have shown that:
Would you like to explore specific steps in more detail or have any further questions?
Relative Questions:
- How can substitution of variables simplify algebraic proofs?
- What are some alternative ways to simplify rational expressions?
- How does the condition affect symmetry in the problem?
- Can this method be generalized for higher-degree polynomial equations?
- What role does factorization play in solving algebraic identities like this?
Tip:
When dealing with symmetrical equations like this, substituting variables based on known conditions (like ) often simplifies the problem significantly.
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Math Problem Analysis
Mathematical Concepts
Algebra
Symmetry in Equations
Rational Expressions
Formulas
Substitution of c = -(a + b)
Rational expression simplification
Theorems
Symmetry and Substitution Theorem
Factorization in Rational Equations
Suitable Grade Level
Grades 11-12
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