Question

$3^{2x+1} -10.3^{x} +3=0$
then x is equal to

Answer 1

Answer :

The value of x are -1 and 1

Given :

The equation is :

$\sf3 {}^{2x + 1} - 10. 3^{x} + 3 = 0$

To Find :

• The value of x

Solution :

$\sf3 {}^{2x + 1} - 10. 3^{x} + 3 = 0 \\ \\ \implies \sf 3 {}^{2x} .3 - 10 .3{}^{x} + 3 = 0 \\ \\ \sf \implies3.{(3 {}^{x} )}^{2} - 10. {3}^{x} + 3 = 0$

$\sf{Let \: us \: consider \: the \: value \: of \: \bold{3 {}^{x} } }\\ \sf be \: \bold{k }\:$

$\sf \implies3k {}^{2} - 10k + 3 = 0 \\ \\ \sf \implies3 {k}^{2} - 9k - k + 3 = 0 \\ \\ \sf \implies3k(k - 3) - 1(k - 3) = 0 \\ \\ \sf \implies(k - 3)(3k - 1) = 0$

Now we have from our consideration:

$\sf \implies k - 3 = 0 \\ \\ \implies \sf k = 3 \\ \\ \sf \implies {3}^{x} = 3 \\ \\ \sf \implies {3}^{x} = {3}^{1} \\ \\ \bf \implies x = 1$

and ,

$\sf \implies 3k - 1 = 0 \\ \\ \sf \implies3k = 1 \\ \\ \sf \implies k = \dfrac{1}{3} \\ \\ \sf \implies {3}^{x} = {3}^{ - 1} \\ \\ \bf \implies x = - 1$

Answer 2

...

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QuEsTi0N -

$3^{2x+1} -10.3^{x} +3=0$

then x is equal to ?

S0LUTI0N -

$3^{2x+1} -10.3^{x} +3=0 \\ \\ Let \: { 3 } ^ x \: be \: a. \\ \\ => 3 { a } ^ 2 - 10 a + 3 = 0 \\ \\ => 3 { a }^ 2 - 9 a - a + 3 = 0 \\ \\ => 3a ( a - 3 ) - 1 ( a - 3 ) = 0 \\ \\ => ( 3a - 1 )( a - 3 ) = 0 \\ \\ => a = \dfrac{1}{3} \: or \: 3 . \\ \\ => { 3 } ^ x = { 3 } ^ { -1 } \\ \\ => x = -1 \\ \\ { 3 } ^ x = { 3 } ^ { 1 } \\ \\ => x = 1$