Question

4.2^2x-6^x=18.3^2x solve for x real no.

Answer 1

Given,
4.2²ˣ - 6ˣ = 18.3²ˣ
4.(2ˣ)² - (2ˣ)(3ˣ) = 18.(3ˣ)²
Let 2ˣ = P and 3ˣ = Q
Then, 4(2ˣ)² - (2ˣ)(3ˣ) - 18(3ˣ)² = 4P² - PQ - 18q² =0
⇒4P² + 8PQ - 9PQ - 18Q² = 0
⇒4P(P + 2Q) - 9Q(P + 2Q) = 0
⇒(4P - 9Q)(P + 2Q) = 0
⇒(4P - 9Q) = 0 and (P + 2Q) = 0

Case 1 :- 4P = 9Q
putting P = 2ˣ and Q = 3ˣ
⇒4.2ˣ = 9.3ˣ
⇒(2)².2ˣ = (3)²3ˣ
⇒2ˣ⁺² = 3ˣ⁺²
⇒(2/3)ˣ⁺² = 1 = (2/3)⁰
⇒x + 2 = 0 ⇒x = -2

Case 2 :- P = -2Q
Putting P = 2ˣ and Q = 3ˣ
2ˣ = -2.3ˣ
But we know, exponential function can't be negative so,
2ˣ ≠ -2.3ˣ


Hence, x = -2 is the correct answer.

Answer 2

HELLO DEAR,

4.2²ˣ - 6ˣ = 18.3²ˣ

4.(2ˣ)² - (2ˣ)(3ˣ) = 18.(3ˣ)²

let [(2ˣ) = a , (3ˣ) = b]

⇒4a² - ab = 18.b²

⇒4a² - 9ab + 8ab - 18b² = 0

⇒a(4a - 9b) + 2b(4a - 9b) = 0

⇒(a + 2b)(4a - 9b) = 0

⇒ (a + 2b) = 0 , (4a - 9b) = 0

Now, let a = -2b

Put [(2ˣ) = a , (3ˣ) = b]

[(2ˣ) = -2(3ˣ)

∴ [ exponential function can't be negative so, (2ˣ) ≠ -2.(3ˣ) ]

Now, let 4a = 9b

Put [(2ˣ) = a , (3ˣ) = b]

4(2ˣ) = 9(3ˣ)

⇒(2²)(2ˣ) = (3²)(3ˣ)

⇒2⁽² ⁺ ˣ⁾ = 3⁽² ⁺ ˣ)

⇒(2/3)⁽² ⁺ ˣ) = 1

⇒(2/3)⁽² ⁺ ˣ) = (2/3)⁰ = 1

⇒(2 + x) = 0

⇒x = -2

I HOPE ITS HELP YOU DEAR,

THANKS