Math Problem Statement
a^3 + b^3 =10 a^2 + b^2 = 7
find a + b
Solution
We have the system of equations:
Let's solve for .
Step 1: Express the identity
The identity for the sum of cubes is: We know from the second equation that: Thus, we can write: Let's set and . Then: And for the sum of cubes:
Step 2: Solve for using Equation 1
Step 3: Substitute into Equation 2
Substitute into the second equation: Multiply through to eliminate the fraction:
Step 4: Solve the cubic equation
By trial, synthetic division, or factoring, the solution to this cubic equation is .
Final Answer
Would you like more details, or do you have any questions?
Here are five related questions for practice:
- If , what are the possible values of ?
- How would the solution change if ?
- Can you find the individual values of and ?
- What is the relationship between and given the solution ?
- How does the discriminant of a cubic equation help in finding the roots?
Tip: For solving cubic equations, trying simple integers as possible roots can often simplify the process significantly.
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Math Problem Analysis
Mathematical Concepts
Cubic Equations
Sum of Cubes
Quadratic Equations
Formulas
Sum of Cubes identity
Quadratic equation formula
Theorems
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Suitable Grade Level
Grades 10-12
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