Question

If (3x-2/x)=5 then find the value of (9x^2-4/x^2)

Answer 1

3x - 2/x = 5 (Given)

Multiply by x through:

3x² - 2 = 5x

Subtract 5x from both sides:

3x² - 5x - 2 = 0

Factorise:

(x - 2)(3x + 1) = 0

Apply zero product property:

x = 2 or x = -1/3

When x = 2

9x² - 4/x² = 9(2)² - 4/(2)²

9x² - 4/x² = 9(4) - 4/4

9x² - 4/x² = 36 - 1

9x² - 4/x² = 35

When x = - 1/3

9x² - 4/x² = 9 (-1/3)² - 4/(-1/3)²

9x² - 4/x² = 9(1/9) - 4/(1/9)

9x² - 4/x² = 1 - 36

9x² - 4/x² = -35

Answer: x = 35 or x = -35

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ALTERNATIVE SOLUTION:

3x - 2/x = 5 (Given)

Square both sides:

(3x - 2/x)² = 5²

Expand (a - b)² = a² - 2ab + b²

(3x)² - 2(3x)(2/x) + (2/x)² = 25

9x² - 12 + 4/x² = 25

Add 12 to both sides:

9x² + 4/x² = 37

Rewrite equation into a² + b²:

(3x)² + (2/x)² = 37

*Side note:

(a + b)² = a² + 2ab + b²

⇒ a² + b² = (a + b)² - 2ab

Apply a² + b² = (a + b)² - 2ab:

(3x)² + (2/x)² = 37

(3x + 2/x)² - 2(3x)(2/x) = 37

(3x + 2/x)² - 12 = 37

(3x + 2/x)² = 49

Square root both sides:

(3x + 2/x)² = 49

(3x + 2/x) = 7

Find 9x² - 4/x² :

9x² - 4/x² = (3x)² - (2/x)²

9x² - 4/x² = (3x + 2/x) (3x - 2/x)

Since we found out that 3x + 2/x = 7 and 3x - 2/x = 5:

9x² - 4/x² = 7 x 5

9x² - 4/x² = 35

Answer: 35