Question

If x + 2y – 5 = 0, then prove that x³ + 8y³ + 30xy = 125.

Answer 1

Given: x + 2y – 5 = 0

To Prove: x³ + 8y³ + 30xy = 125

Proof:

Step 1: Rewrite the given equation as:

x + 2y = 5

Step 2: Cube both sides:

(x + 2y)³ = 5³

Step 3: Expand the left-hand side using the binomial expansion:

(x + 2y)³ = x³ + 3x²(2y) + 3x(2y)² + (2y)³

(x + 2y)³ = x³ + 6x²y + 12xy² + 8y³

Step 4: Group terms:

x³ + 8y³ + 6x²y + 12xy² = 125

Step 5: Factor the middle terms:

6x²y + 12xy² = 6xy(x + 2y)

Step 6: Substitute x + 2y = 5 into the expression:

x³ + 8y³ + 6xy × 5 = 125

Step 7: Simplify:

x³ + 8y³ + 30xy = 125

Hence, x³ + 8y³ + 30xy = 125.