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Solution:
[tex](i) \: 2 {}^{2x + 1} = 17. {2}^{x} - 2 {}^{3} [/tex]
[tex] \implies \: 2.2 {}^{2x} = 17.2 {}^{x} - 2 {}^{3} [/tex]
[tex] \implies \: 2.(2 { }^{x} ) {}^{2} = 17.( {2}^{x} ) - 8[/tex]
Now,
[tex]Let \: 2 {}^{x} = a[/tex]
[tex] \implies \: 2a {}^{2} = 17a - 8 [/tex]
[tex] \implies \: 2 {a}^{2} - 17a + 8 = 0[/tex]
[tex] \implies \: 2a {}^{2} - 16a - a + 8 = 0[/tex]
[tex] \implies \: 2a(a - 8) - 1(a - 8) = 0[/tex]
[tex] \implies \: (a- 8)(2a - 1) = 0[/tex]
[tex] \implies \: a = 8 \: and \: a = \frac{1}{2} [/tex]
[tex] \star \: {2}^{x} = 8 \implies {2}^{x} = {2}^{3} \implies \: x = 3[/tex]
[tex] \star \: {2}^{x }= \frac{1}{2} \implies {2}^{x} = {2}^{ - 1} \implies x = - 1[/tex]
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[tex](ii) \: 5 {}^{2x + 1} = 6.5x - 1[/tex]
[tex] \implies 5.5 {}^{2x} = 6.(5 {}^{x} ) - 1[/tex]
[tex] \implies \: 5( {5}^{x} ) {}^{2} = 6(5 {}^{x} ) - 1[/tex]
Now,
[tex]Let \: {5}^{x} = a[/tex]
[tex] \implies \: 5a {}^{2} - 6a + 1 = 0[/tex]
[tex] \implies \: 5a {}^{2} - 5a - a + 1 = 0[/tex]
[tex] \implies \: 5a(a - 1) - 1(a - 1) = 0[/tex]
[tex] \implies \: (5a - 1)(a - 1) = 0[/tex]
[tex] \implies \: a = \frac{1}{5} \: and \: a = 1 [/tex]
[tex] \star \: 5 {}^{x} = \frac{1}{5} \implies5 {}^{x} = 5 {}^{ - 1} \implies \: x = - 1[/tex]
[tex] \star {5}^{x} = 1 \implies {5}^{x} = {5}^{0 } \implies \: x = 0[/tex]
Step-by-step explanation:
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