Algebra problem. Solve 2^x - 6(2^-x) = 6?

Question

I think you use U-substitution but I'm not quite sure how to work it. Could somebody show me this problem step by step?
And no this is not a homework question, I'm studying for my final that I have tomorrow. Thanks in advance to all.

That's where I got too also marley with u substitution but then i tried using the quadratic formula to figure out u and it comes out to a weird answer, if somebody could figure out "x" that would be very helpful.

Answer 1

2^x - 6(2^-x) = 6

Let u = 2^x
then u - 6/u = 6

Multiply both sides by u:
u^2 - 6 = 6u

Subtract 6u from both sides:
u^2 - 6u - 6 = 0

Use quadratic formula to factor:
u = [6 ± sqrt(36 + 24)] / 2
u = [6 ± sqrt(60)] / 2
u = [6 ± 2sqrt(15)] / 2
u = 3 ± sqrt(15)

So
2^x = 3 ± sqrt(15)

Take the ln of both sides:
ln(2^x) = ln(3 ± sqrt(15))
Since you can only take the log of a positive number, this eliminates the negative solution.
x * ln(2) = ln(3 + sqrt(15))
x = ln(3 + sqrt(15)) / ln(2)

So x is approximately 2.781. If you plug this into the original equation, you will see that the answer checks.


Good luck with your final.

Answer 2

I read this as
2 to the power x minus 6 times (2 to the power minus x) equals 6.

2^x - 6/(2^x) = 6

Put everything over the same denominator (2^x)

[(2^x)^2 - 6] / 2^x = 6(2^x) / 2^x

Since the denominators are the same on both sides, the numerators must be equal

(2^x)^2 - 6 = 6(2^x)
1(2^x)^2 - 6(2^x) - 6 = 0

If you want to do this by substitution, try y = 2^x

y^2 - 6y - 6 = 0

and use the quadratic formula to find y.

Once you have one or two values for y, take one and make it equal to 2^x

2^x = y
take the logarithm in base 2 on both sides:
Log_2(2^x) = Log_2(y)
x = Log_2(y)

hint
Log_2(y) = Log_10(y) / Log_10(2)
or
Log_2(y) = ln(y)/ln(2) (natural logs)

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whitesox08's answer is clearer than mine.

Answer 3

First, rewrite it without the negative exponent:

2^x - 6/2^x - 6 = 0

multiply through by 2^x:

2^(2x) - 6 -6(2^x) = 0

Let u = 2^x

u^2 -6u - 6 = 0

I got it to here -- but it doesn't factor :(
Well, I hope this much helps. Good luck with your final!

Answer 4

2^x - 6/2^x - 6 = 0

multiply through by 2^x:

2^(2x) - 6 -6(2^x) = 0

Let u = 2^x

u^2 -6u - 6 = 0

Answer 5

ok so the 1st element you wanna do is discover the distributive sources: 12x-80 4 + 6x+6 = 50x-sixty two then you definately wanna simplify the left component to the equation: 18x-seventy 8 = 50x-sixty two then you definately subtract 18x from the two sides: seventy 8=50x-one hundred forty then you definately upload one hundred forty to the two sides: 218=50x and then finally you divide the two sides by using 50: 4.36=x (i will have executed the somewhat math incorrect, yet this is unquestionably the way you resolve it)