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What is the sum of all solutions of 49^x + 7^(1-2x) = 8

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Bunuel
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What is the sum of all solutions of 49^x + 7^(1-2x) = 8 [#permalink] New post   03 Dec 2020, 01:15
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What is the sum of all solutions of \(49^x + 7^{(1-2x)} = 8\) ?

A. -1
B. -1/2
C. 0
D. 1/2
E. 1


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D
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FauleKatze
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Re: What is the sum of all solutions of 49^x + 7^(1-2x) = 8 [#permalink] New post   03 Dec 2020, 07:38
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Bunuel wrote:
What is the sum of all solutions of \(49^x + 7^{(1-2x)} = 8\) ?

A. -1
B. -1/2
C. 0
D. 1/2
E. 1


\(49^x + 7^{(1-2x)} = 8\) can be written as: \(7^{2x} + 7^{(1-2x)} = 7^1 + 7^0\)
=> Comparing the 2 parts, 2x=1 or 1-2x =1
=> x=1/2 or 0

Henceforth, answer : D
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Re: What is the sum of all solutions of 49^x + 7^(1-2x) = 8 [#permalink] New post   03 Dec 2020, 20:44
2
Kudos
IMO D

49^x+7^(1−2x)=8
=>7^(2x)+7^(1−2x)=\(7^1\)+\(7^0\)

Therefore possible values, are
2x=1 and 1-2x = 0
=> x = 1/2

or
2x = 0 and 1-2x = 1
=> x = 0

adding the two values = 1/2+0 = 1/2
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Re: What is the sum of all solutions of 49^x + 7^(1-2x) = 8 [#permalink] New post   03 Dec 2020, 03:32
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49^x + 7^(1-2x) = 8
or, 7^(2x) + 7^(1-2x) = 8
or, 7^(2x) + 7/7^(2x) = 8
or, a+7/a =8 [ let, 7^(2x) =a ]
or, a^2-8a+7=0
or, a=1, 7
so, 7^(2x) =1 =7^0
or,x=0
and, 7^(2x)=7 = 7^1
or, x=1/2

therefore, the sum of all solutions of 49^x + 7^(1-2x) = 8 is (0+1/2) =1/2

correct answer is D
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Re: What is the sum of all solutions of 49^x + 7^(1-2x) = 8 [#permalink] New post   03 Dec 2020, 05:27
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IMO D
Take 49^x = y
=> quadratic equation in terms of y. This has roots: 7 and 1.
=> 49^x = 1 OR 49^x = 7
=> x=0 OR x=1/2
Sum of roots= 0 + 1/2 = 1/2



Bunuel wrote:
What is the sum of all solutions of \(49^x + 7^{(1-2x)} = 8\) ?

A. -1
B. -1/2
C. 0
D. 1/2
E. 1


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Re: What is the sum of all solutions of 49^x + 7^(1-2x) = 8 [#permalink] New post   03 Dec 2020, 15:01
1
Kudos
Bunuel wrote:
What is the sum of all solutions of \(49^x + 7^{(1-2x)} = 8\) ?

A. -1
B. -1/2
C. 0
D. 1/2
E. 1


7^2x+7/7^2x=8
a^2+7-8a=0
(a-7)(a-1)=0
7^2x=7, x=1/2
7^2x=1, x=0
(D)
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Re: What is the sum of all solutions of 49^x + 7^(1-2x) = 8 [#permalink]
03 Dec 2020, 15:01
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