Math Problem Statement

a^3 + b^3 =10 a^2 + b^2 = 7

find a + b

Solution

We have the system of equations:

a3+b3=10a^3 + b^3 = 10 a2+b2=7a^2 + b^2 = 7

Let's solve for a+ba + b.

Step 1: Express the identity

The identity for the sum of cubes is: a3+b3=(a+b)(a2ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) We know from the second equation that: a2+b2=7a^2 + b^2 = 7 Thus, we can write: a2+b2=(a+b)22ab=7a^2 + b^2 = (a + b)^2 - 2ab = 7 Let's set s=a+bs = a + b and p=abp = ab. Then: s22p=7(Equation 1)s^2 - 2p = 7 \quad \text{(Equation 1)} And for the sum of cubes: a3+b3=s(s23p)=10(Equation 2)a^3 + b^3 = s(s^2 - 3p) = 10 \quad \text{(Equation 2)}

Step 2: Solve for pp using Equation 1

p=s272p = \frac{s^2 - 7}{2}

Step 3: Substitute pp into Equation 2

Substitute pp into the second equation: s(s23p)=10s(s^2 - 3p) = 10 s(s23(s272))=10s\left(s^2 - 3\left(\frac{s^2 - 7}{2}\right)\right) = 10 Multiply through to eliminate the fraction: s(s232(s27))=10s\left(s^2 - \frac{3}{2}(s^2 - 7)\right) = 10 s(s23s2212)=10s\left(s^2 - \frac{3s^2 - 21}{2}\right) = 10 s(2s23s2+212)=10s\left(\frac{2s^2 - 3s^2 + 21}{2}\right) = 10 s(s2+212)=10s\left(\frac{-s^2 + 21}{2}\right) = 10 s(s2+21)=20s(-s^2 + 21) = 20 s3+21s=20-s^3 + 21s = 20

Step 4: Solve the cubic equation

s321s+20=0s^3 - 21s + 20 = 0

By trial, synthetic division, or factoring, the solution to this cubic equation is s=1s = 1.

Final Answer

a+b=1a + b = 1

Would you like more details, or do you have any questions?


Here are five related questions for practice:

  1. If a+b=1a + b = 1, what are the possible values of abab?
  2. How would the solution change if a3+b3=15a^3 + b^3 = 15?
  3. Can you find the individual values of aa and bb?
  4. What is the relationship between aa and bb given the solution a+b=1a + b = 1?
  5. 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