Question

If $x + y = 7$ and $xy = 10$, find: $(x^3 + y^3)^2 - 9x^2y^2(x + y)^2$

• A. 24011
• B. - 26411
• C. 12255
• D. – 12251

Answer 1

Answer:

B. - 26411

Step-by-Step Algebraic Solution

• Objective: Calculate the value of (x³ + y³)² - 9x²y²(x + y)² given x + y = 7 and xy = 10.

• Formula 1 (Sum of Cubes): Recall that x³ + y³ = (x + y)(x² - xy + y²). Alternatively, use the identity x³ + y³ = (x + y)³ - 3xy(x + y).

• Calculation of x³ + y³: Using the identity (x + y)³ - 3xy(x + y):
  Substitute x + y = 7 and xy = 10:
  (7)³ - 3(10)(7) = 343 - 210 = 133.

• Substitute values into the expression: The original expression is (x³ + y³)² - 9x²y²(x + y)².
  We know x³ + y³ = 133, xy = 10, and x + y = 7.

• Evaluate terms:
  Term 1: (133)² = 17689.
  Term 2: 9x²y²(x + y)² = 9(xy)²(x + y)².
  Substitute xy = 10 and x + y = 7:
  9(10)²(7)² = 9(100)(49) = 9(4900) = 44100.

• Final Calculation: 17689 - 44100 = -26411.

Key Algebraic Identities for Competitive Exams

• Perfect Cube Identity: (x + y)³ = x³ + y³ + 3xy(x + y). This is crucial for solving problems involving sums of cubes.

• Square of a Sum: (x + y)² = x² + 2xy + y².

• Useful Simplification Tip: In competitive exams, always check if you can simplify the expression algebraically before substituting numerical values. For example, the given expression (x³ + y³)² - 9x²y²(x + y)² can be factored as a difference of squares if rewritten, though direct substitution is often faster for these specific values.