find the absolute value of the difference of the solutions to the equation 3^2x -10(3^x) +9=0.

Question

turingtest · Answer

let u=3^x

anonymous · Answer

@TuringTest then what would the equation be? and what would you do next?

turingtest · Answer

if u=3^x what is u^2=?

anonymous · Answer

@TuringTest would it be 3^2x?

turingtest · Answer

yep :)
so now make the substitutions into the equation and you will get a regular quadratic

anonymous · Answer

@TuringTest the the equation would be U^2-10U+9 and then you factor? i have no clue!

turingtest · Answer

what two numbers multiply to make 9 and add to make -10 ?

anonymous · Answer

@TuringTest i know how to factor and i got (u-9)(u-1) then what

turingtest · Answer

the zero factor property states that$$(u-9)(u-1)=0$$implies that either$$u-9=0~~~~or~~~~u-1=0$$or both
solve those two equations separately, then sub back in for u

anonymous · Answer

@TuringTest can you show me how to do that im confused

turingtest · Answer

can you solve u-9=0 for u ? I sure hope so

anonymous · Answer

yes so you put those answers back in to the equation for X then what
@TuringTest

turingtest · Answer

what do you mean "put them in the equation for x" ? We are solving for u, not x yet
what are the two solutions for u ?

anonymous · Answer

9 and 1
@TuringTest

turingtest · Answer

yes, so we have u=1 and u=9
now sub back in our original expression for u and what do we get?

anonymous · Answer

would the equations then be 3^18 -10(3^9)+9 and 3^2 -10(3) +9
@TuringTest

turingtest · Answer

you plugged in the values for u into where the x goes, that's not right.
all I'm asking you to do is put the expressions u=1 and u=9 back in terms of x

turingtest · Answer

you also don't need to write @turingtest every time...

turingtest · Answer

they want to absolute value of the difference of the two solutions
we have the two solutions in terms of u
undo our original sub for u to find what x is supposed to be

anonymous · Answer

9=3^x
1=3^x?

turingtest · Answer

yes, excellent
can you solve those for x?

anonymous · Answer

x=2,0

turingtest · Answer

great, and the absolute value of the difference of those two solutions is...?

anonymous · Answer

2

turingtest · Answer

which is your answer, nice job :)

anonymous · Answer

thanks so much!:)

turingtest · Answer

very welcome!