How do I slove this quadratic equation?

[kboynt1]

Homework Statement

I'm studying for the GMAT and the study materials do not include how to solve this equation. I need to know how to solve it without a calculator.

Homework Equations

If 2^x - 2^x-2 = 3(2^13), what is the value of X?

The Attempt at a Solution

The answer key says the the answer is 15. How did they come to this?

 

[tiny-tim]

welcome to pf!

hi kboynt1! welcome to pf!

(try using the X² button just above the Reply box )

do you mean 2^x - 2^x-2 ?

hint: what is 2^x/2^x-2 ? ​

 

[kboynt1]

Would it be 2^x-(x-2)?

 

[tiny-tim]

sure would!

and that equals … ? ​

 

[kboynt1]

Holy Poop tim! my brain is about to explode... I don't know. :)

 

[tiny-tim]

ok, when you're confused, do it one logical step at a time …

what is x - (x - 2) ? ​

 

[kboynt1]

wait... is it 2^2x-2?

 

[kboynt1]

or is it... 2^x/x-2?

 

[kboynt1]

so we have to simplify the left side of the equation first?

2^x-2^x-2=3(2¹³)

Then

2^x-(x-2)=3(2¹³)

Then...?

 

[kboynt1]

ok, when you're confused, do it one logical step at a time …

what is x - (x - 2) ? ​

is it just 2?

so it's 2²=4

so

4=3(2¹³)? That doesn't seem right...

 

[tiny-tim]

is it just 2?

so it's 2²=4

yes!

now stop trying to do both side of the equation at once

just do the LHS, and keep changing it until it matches the RHS

you have 2^x/2^x-2 = 4

so (2^x - 2^x-2)/2^x-2 = … ?

 

[kboynt1]

is it?

2^x-(x-2)-2^x-2-(x-2)

4-1=3

 

[tiny-tim]

is it?

2^x-(x-2)-2^x-2-(x-2)

4-1=3

yup!

so your LHS is now 3(2^x-2), and your RHS is 3(2¹³),

soooo … ? ​

 

[kboynt1]

3(2^x-2)=3(2¹³)

divide by 3 on both sides

2^x-2=2¹³

add 2 to both the exponents

2^x=2¹⁵

x must equal 15.


Okay... so I'm still unclear of how you got to 3(2^x-2) on the LHS I know how we got the 3 but what were the thoughts behind that and how did we retain the (2^x-2)

THANK YOU BTW... Just a little more explanation/breakdown.

 

[tiny-tim]

2^x - 2^x-2

= 2²2^{x - 2} - 2^x-2

= (2² - 1)2^x-2

 

[kboynt1]

2^x - 2^x-2

= 2²2^{x - 2} - 2^x-2

= (2² - 1)2^x-2

So what's the rule that governs the division that was done to obtain the 3 but retain the 2^x-1

Sorry I'm a bit dense on this subject...

for example is the rule: x^y-x^z=((x^y-x^z)/x^z)x^z

 

[tiny-tim]

the rule is AB - B = (A - 1)B