Find the value of a + b given the equation a³ + b³ = 1 + 3ab | Step-by-Step Solution

1. What We're Solving:

We need to find the value of a + b when we know that a³ + b³ = 1 + 3ab.

2. The Approach:

This problem is asking us to recognize a special algebraic identity! The key insight is that the equation looks very similar to the factorization formula for the sum of cubes. We'll use the identity a³ + b³ = (a + b)(a² - ab + b²) and some clever algebraic manipulation to solve this.

3. Step-by-Step Solution:

Step 1: Start with the sum of cubes identity We know that: a³ + b³ = (a + b)(a² - ab + b²)

Step 2: Substitute this into our original equation Since a³ + b³ = 1 + 3ab, we can write: (a + b)(a² - ab + b²) = 1 + 3ab

Step 3: Look for a pattern with a² - ab + b² a² - ab + b² = a² + 2ab + b² - 3ab = (a + b)² - 3ab

Step 4: Substitute this back into our equation (a + b)[(a + b)² - 3ab] = 1 + 3ab

Step 5: Let's make a substitution to simplify Let s = a + b. Then our equation becomes: s(s² - 3ab) = 1 + 3ab

Step 6: Expand and rearrange s³ - 3abs = 1 + 3ab s³ = 1 + 3ab + 3abs s³ = 1 + 3ab(1 + s)

Step 7: Try s = 1 as a solution Let's test if s = 1 (meaning a + b = 1) works: If s = 1, then: 1³ = 1 + 3ab(1 + 1) 1 = 1 + 6ab 0 = 6ab Therefore ab = 0

Step 8: Verify our solution When a + b = 1 and ab = 0, let's check our original equation:

• If ab = 0, then either a = 0 or b = 0
• If a = 0, then b = 1: 0³ + 1³ = 1 and 1 + 3(0)(1) = 1 ✓
• If b = 0, then a = 1: 1³ + 0³ = 1 and 1 + 3(1)(0) = 1 ✓

4. The Answer:

a + b = 1

5. Memory Tip:

Remember that sum of cubes problems often involve the identity a³ + b³ = (a + b)(a² - ab + b²). When you see a³ + b³ in an equation, try factoring it first! Also, the relationship (a + b)² = a² + 2ab + b² is your friend for connecting a² + b² terms with (a + b).

Great job working through this! These types of problems reward pattern recognition and knowing your algebraic identities. Keep practicing, and you'll start spotting these opportunities more quickly! 🌟