2^2x+3+1=3^22^x 2^2x2^3+1 9.2^x=0 8.2^2x+1 9.2^x=0 Let 2^x=t⇒2^2x=t^2 8t^2 9t+1=0 ⇒ 8t^2 8t t+1=0 ⇒ 8t(t 1) (t 1)=0 ⇒(t ...

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Comparision of Quantities Using Exponents

Solve: 22x+3...

Question

Solve: $2^{2 x + 3} + 1 = 3^{2} 2^{x}$

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Solution

$2^{2 x + 3} + 1 = 3^{2} 2^{x}$

$2^{2 x} 2^{3} + 1 - 9. 2^{x} = 0$

$8. 2^{2 x} + 1 - 9. 2^{x} = 0$

Let $2^{x} = t \Rightarrow 2^{2 x} = t^{2}$

$8 t^{2} - 9 t + 1 = 0$

$\Rightarrow 8 t^{2} - 8 t - t + 1 = 0$

$\Rightarrow 8 t (t - 1) - (t - 1) = 0$

$\Rightarrow (t - 1) (8 t - 1) = 0$

$\Rightarrow t = 1, t = \frac{1}{8} = \frac{1}{2^{3}} = 2^{- 3}$

$\Rightarrow 2^{x} = 1, 2^{x} = 2^{- 3}$

$\Rightarrow 2^{x} = 2^{0}, 2^{x} = 2^{- 3}$

Since bases are the same we can equate the powers.

$∴ x = 0, - 3$

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