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Answer 1

Solution:

$(i) \: 2 {}^{2x + 1} = 17. {2}^{x} - 2 {}^{3}$

$\implies \: 2.2 {}^{2x} = 17.2 {}^{x} - 2 {}^{3}$

$\implies \: 2.(2 { }^{x} ) {}^{2} = 17.( {2}^{x} ) - 8$

Now,

$Let \: 2 {}^{x} = a$

$\implies \: 2a {}^{2} = 17a - 8$

$\implies \: 2 {a}^{2} - 17a + 8 = 0$

$\implies \: 2a {}^{2} - 16a - a + 8 = 0$

$\implies \: 2a(a - 8) - 1(a - 8) = 0$

$\implies \: (a- 8)(2a - 1) = 0$

$\implies \: a = 8 \: and \: a = \frac{1}{2}$

$\star \: {2}^{x} = 8 \implies {2}^{x} = {2}^{3} \implies \: x = 3$

$\star \: {2}^{x }= \frac{1}{2} \implies {2}^{x} = {2}^{ - 1} \implies x = - 1$

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$(ii) \: 5 {}^{2x + 1} = 6.5x - 1$

$\implies 5.5 {}^{2x} = 6.(5 {}^{x} ) - 1$

$\implies \: 5( {5}^{x} ) {}^{2} = 6(5 {}^{x} ) - 1$

Now,

$Let \: {5}^{x} = a$

$\implies \: 5a {}^{2} - 6a + 1 = 0$

$\implies \: 5a {}^{2} - 5a - a + 1 = 0$

$\implies \: 5a(a - 1) - 1(a - 1) = 0$

$\implies \: (5a - 1)(a - 1) = 0$

$\implies \: a = \frac{1}{5} \: and \: a = 1$

$\star \: 5 {}^{x} = \frac{1}{5} \implies5 {}^{x} = 5 {}^{ - 1} \implies \: x = - 1$

$\star {5}^{x} = 1 \implies {5}^{x} = {5}^{0 } \implies \: x = 0$

Answer 2

Step-by-step explanation:

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