a² + b² Formulas: Complete guide to all variations explained

Extended Three-Variable Formulas

Specialized Formulas and Variations

Detailed Formula Explanations

1. a² + b² = (a + b)² – 2ab

Proof:

• Start with (a + b)² = a² + 2ab + b²
• Rearranging: a² + b² = (a + b)² – 2ab

Application: Finding sum of squares when sum and product are known.

2. a² – b² = (a + b)(a – b)

Proof:

• Expand (a + b)(a – b)
• = a(a – b) + b(a – b)
• = a² – ab + ab – b²
• = a² – b²

Application: Factorization, simplification of algebraic expressions.

3. (a + b)² = a² + 2ab + b²

Proof:

• (a + b)(a + b)
• = a(a + b) + b(a + b)
• = a² + ab + ab + b²
• = a² + 2ab + b²

Application: Expanding binomial squares, solving quadratic equations.

4. (a – b)² = a² – 2ab + b²

Proof:

• (a – b)(a – b)
• = a(a – b) – b(a – b)
• = a² – ab – ab + b²
• = a² – 2ab + b²

Application: Completing the square, algebraic simplifications.

5. 2(a² + b²) = (a + b)² + (a – b)²

Proof:

• (a + b)² + (a – b)²
• = (a² + 2ab + b²) + (a² – 2ab + b²)
• = 2a² + 2b²
• = 2(a² + b²)

Application: Geometric mean problems, optimization.

6. a² + b² + c² = (a + b + c)² – 2(ab + bc + ca)

Proof:

• Expand (a + b + c)²
• = a² + b² + c² + 2ab + 2bc + 2ca
• Rearranging: a² + b² + c² = (a + b + c)² – 2(ab + bc + ca)

Application: Three-variable algebraic problems.

7. a² + b² + c² – ab – bc – ca = ½[(a – b)² + (b – c)² + (c – a)²]

Proof:

• Expand right side:
• ½[(a² – 2ab + b²) + (b² – 2bc + c²) + (c² – 2ca + a²)]
• = ½[2a² + 2b² + 2c² – 2ab – 2bc – 2ca]
• = a² + b² + c² – ab – bc – ca

Application: Proving inequalities, optimization problems.

Frequently Asked Questions about a² + b² formula

Q. What is the a² + b² formula and can it be factored?

The expression a² + b² represents the sum of two squares and cannot be factored using real numbers. However, it can be expressed as a² + b² = (a + b)² – 2ab or factored using complex numbers as (a + bi)(a – bi). Unlike a² – b², which factors to (a + b)(a – b), the sum of squares has no real factorization.

Q. How do you prove the a² + b² = (a + b)² – 2ab formula?

Proof: Start with the expansion of (a + b)²:

• (a + b)² = a² + 2ab + b²
• Subtract 2ab from both sides: a² + b² = (a + b)² – 2ab

This identity is particularly useful when you know the sum (a + b) and product (ab) of two numbers but need to find a² + b².

Q. What is the difference between (a + b)² and a² + b²?

• (a + b)² = a² + 2ab + b² (includes the cross term 2ab)
• a² + b² = a² + b² (no cross term)

The key difference is the middle term 2ab. Students often mistakenly write (a + b)² = a² + b², which is incorrect. The correct formula always includes the cross-product term.

Q. What is the a² + b² = c² formula called?

This is the Pythagorean Theorem, one of the most fundamental formulas in geometry. It states that in a right-angled triangle, the square of the hypotenuse (c) equals the sum of squares of the other two sides (a and b). This formula has countless applications in mathematics, physics, engineering, and navigation.

Q. How is the formula 2(a² + b²) related to (a + b)² and (a – b)²?

The relationship is: 2(a² + b²) = (a + b)² + (a – b)²

Proof:

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• Adding: (a + b)² + (a – b)² = 2a² + 2b² = 2(a² + b²)

This formula is useful in solving optimization and geometry problems.

Q. What is the a² + b² + c² formula expansion?

The main formula is: a² + b² + c² = (a + b + c)² – 2(ab + bc + ca)

Derivation:

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• Rearranging: a² + b² + c² = (a + b + c)² – 2(ab + bc + ca)

This is essential for problems involving three variables.

Q. How do you use a² – b² = (a + b)(a – b) in problem-solving?

The difference of squares formula is used for:

• Factorization: 49 – 16 = 7² – 4² = (7 + 4)(7 – 4) = 11 × 3 = 33
• Simplification: (x² – 9)/(x – 3) = (x + 3)(x – 3)/(x – 3) = x + 3
• Mental calculation: 103² – 97² = (103 + 97)(103 – 97) = 200 × 6 = 1200

This formula saves time and simplifies complex calculations.

Q. What are the real-world applications of a² + b² formulas?

These formulas are used in:

• Construction & Architecture: Calculating diagonal distances, ensuring right angles
• Navigation: Finding shortest distances (GPS calculations)
• Physics: Calculating resultant vectors, wave mechanics
• Computer Graphics: Distance calculations, 3D modeling
• Engineering: Stress analysis, electrical impedance calculations
• Data Science: Euclidean distance in machine learning algorithms

Q. What is the relationship between ab, bc, ca and a² + b² + c²?

The key formula is: a² + b² + c² – ab – bc – ca = ½[(a – b)² + (b – c)² + (c – a)²]

This proves that a² + b² + c² ≥ ab + bc + ca for all real numbers (equality only when a = b = c). This inequality is fundamental in optimization and proving mathematical theorems.

Q. How do you remember all these a² + b² formulas for exams?

Memory Tips:

1. Practice the derivations – understanding helps retention more than memorization
2. Use mnemonics: “(a + b)² has a PLUS 2ab” and “(a – b)² has a MINUS 2ab”
3. Pattern recognition: Notice symmetry (a² – b² has both + and -, while a² + b² doesn’t factor)
4. Solve 10-15 problems daily using each formula
5. Create a formula sheet and review it before bed
6. Relate to geometry: visualize squares and rectangles for (a + b)²
7. Group formulas: two-variable, three-variable, special identities

Regular practice with varied problem types ensures these formulas become second nature.