How do I slove this quadratic equation?

In summary: A - 1)2x-2=1B so 3 would be the result.In summary, Homework Equations asks for X to be 15. If 2^x - 2^x-2 = 3(2^13), what is the value of X? The attempt at a solution says the answer is 15. How did they come to this? The answer key says the the answer is 15. Do you mean 2x - 2x-2 ? Yes, that's what I meant. And that equals … ? 4. So x must equal 15.
  • #1
kboynt1
10
0

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?
 
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  • #2
welcome to pf!

hi kboynt1! welcome to pf! :smile:

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

do you mean 2x - 2x-2 ?

hint: what is 2x/2x-2 ? :wink:
 
  • #3
Would it be 2x-(x-2)?
 
  • #4
sure would! :smile:

and that equals … ? :wink:
 
  • #5
Holy Poop tim! my brain is about to explode... I don't know. :)
 
  • #6
ok, when you're confused, do it one logical step at a time …

what is x - (x - 2) ? :smile:
 
  • #7
wait... is it 22x-2?
 
  • #8
or is it... 2x/x-2?
 
  • #9
so we have to simplify the left side of the equation first?

2x-2x-2=3(213)

Then

2x-(x-2)=3(213)

Then...?
 
  • #10
tiny-tim said:
ok, when you're confused, do it one logical step at a time …

what is x - (x - 2) ? :smile:

is it just 2?

so it's 22=4

so

4=3(213)? That doesn't seem right...
 
  • #11
kboynt1 said:
is it just 2?

so it's 22=4

yes! :smile:

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 2x/2x-2 = 4

so (2x - 2x-2)/2x-2 = … ? :smile:
 
  • #12
is it?

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

4-1=3
 
  • #13
kboynt1 said:
is it?

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

4-1=3

yup! :smile:

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

soooo … ? :wink:
 
  • #14
3(2x-2)=3(213)

divide by 3 on both sides

2x-2=213

add 2 to both the exponents

2x=215

x must equal 15.


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

THANK YOU BTW... Just a little more explanation/breakdown.
 
  • #15
2x - 2x-2

= 222x - 2 - 2x-2

= (22 - 1)2x-2
 
  • #16
tiny-tim said:
2x - 2x-2

= 222x - 2 - 2x-2

= (22 - 1)2x-2

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

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

for example is the rule: xy-xz=((xy-xz)/xz)xz
 
  • #17
the rule is AB - B = (A - 1)B
 

1. What is a quadratic equation?

A quadratic equation is an equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It represents a parabola on a graph and can have two solutions.

2. How do I identify a quadratic equation?

A quadratic equation will always have a variable raised to the second power. It can also be identified by its standard form, ax^2 + bx + c = 0.

3. How do I solve a quadratic equation using the quadratic formula?

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a. Substitute the values of a, b, and c into the formula and solve for x. The ± symbol means you will have two solutions, one with a plus sign and one with a minus sign.

4. Can I solve a quadratic equation by factoring?

Yes, if the quadratic equation is in the form of ax^2 + bx + c = 0, where a, b, and c are integers, you can solve it by factoring. This method involves finding two numbers that add up to b and multiply to ac, and then using those numbers to rewrite the equation in the form of (x + m)(x + n) = 0. From there, you can solve for x.

5. What if the quadratic equation has imaginary solutions?

If the discriminant (b^2 - 4ac) is negative, the quadratic equation will have imaginary solutions. This means that the solutions will involve the imaginary number i, which is equal to √(-1). The solutions will be in the form of x = (-b ± i√(4ac - b^2)) / 2a.

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