Question

-3^(-x)-6=-3^x+10

Solve the equation below for x by graphing plz

Answer 1

to graph it, just graph
`y=-3^(-x)-6` and
`y=-3^x+10` and see where they intersect

I would like to solve it by using math and not graphing
if you don't want to see the math, just don't scroll down
the graphical meathod is above, first line, just read it

hmm
multiply both sides by -1

`3^(-x)+6=3^x-10`
multiply both sides by
`3^x`

`3^0+6(3^x)=3^(2x)-10(3^x)`

`1+6(3^x)=3^(2x)-10(3^x)`
minus 1 from both sides and minus 6(3^x) from both sides

`0=3^(2x)-16(3^x)-1`
use u subsitution

`u=3^x`
we can rewrite it as

`0=u^2-16u-1`
now factor
I mean use quadratic formula
for
`ax^2+bx+c=0`
`x=(-b+/-√(b^2-4ac))/(2a)`
so for 0=u^2-16u-1, a=1, b=-16, c=-1



`u=8+/-√(65)`
remember that u=3^x so u>0
if we have u=8+√65, it's fine, but u=8-√65 is negative and not allowed
so therfor

`u=8+√(65)=3^x`

`8+√(65)=3^x`
if you take the log base 3 of both sides you get

`log_3(8+√(65))=x`
if you use ln then

`ln(8+√(65))=xln(3)`
then

`(ln(8+√(65)))/(ln(3))=x`

Answer 2

x = 2.527

Explanation:

We have to solve the equation
`-3^(-x)-6=-3^x+10` by graphing method.

In order to solve this equation from graphing method, we graph the below two equations in the same coordinate axes.

`y=-3^(-x)-6...(i)\\\\y=-3^x+10...(ii)`

The x -coordinate of the intersection point of these two graphs would be the solution of the equation.

Please see the attached graph.

The x-coordinate of the intersection point is 2.527.

Hence, the solution to the given equation is x = 2.527