Question

If a - b = 4 and a³ - b³ = 88, then find the value of a² - b².

• A. $$8\sqrt{6}$$
• B. $$6\sqrt{6}$$
• C. $$7\sqrt{2}$$
• D. $9\sqrt{6}$

Answer 1

Answer:

$$8\sqrt{6}$$

Given:

a - b = 4 and a³ - b³ = 88

Formula: 

(a³ - b³) = (a - b) (a² + b² + ab)

(a³ - b³) = (a - b) [(a - b)² + 3ab]

(a + b)² = a² + b² + 2ab

(a + b)(a - b) = a² - b²

Calculation:

According to the formula

88 = 4 $\times$ (4² + 3ab)

⇒ 22 = 16 + 3ab

⇒ 3ab = 22 - 16

⇒ ab = $\frac{6}{3}$

⇒ ab = 2

(a - b)² = a² + b² - 2ab

⇒ 4² = a² + b² - 2 $\times$ 2

⇒ a² + b² = 16 + 4 

⇒ a² + b² = 20

(a + b)² = 20 + 2 $\times$ 2

⇒ a + b = $\sqrt{24} = 2\sqrt{6}$

Then (a + b)(a - b) = $2\sqrt{6}\times{4}$

⇒ $8\sqrt{6}$ 

∴ (a² - b²) = $8\sqrt{6}$