5^x = ( √5 )^x+4 solve it .
Answers
Answer:
x=4
x=-4
Step-by-step explanation:
STEP
1
:
Isolate a square root on the left hand side :
Original equation
√5-x+√5+x = 4
Isolate
√5-x = -√5+x+4
STEP
2
:
Eliminate the radical on the left hand side :
Raise both sides to the second power
(√5-x)2 = (-√5+x+4)2
After squaring
5-x = 5+x+16-8√5+x
STEP
3
:
Get remaining radical by itself :
Current equation
5-x = 5+x+16-8√5+x
Isolate radical on the left hand side
8√5+x = -5+x+5+x+16
Tidy up
8√5+x = 16+2x
STEP
4
:
Eliminate the radical on the left hand side :
Raise both sides to the second power
(8√5+x)2 = (16+2x)2
After squaring
320+64x = 4x2+64x+256
STEP
5
:
Solve the quadratic equation :
Rearranged equation
4x2 -64 = 0
This equation has two rational roots:
{x1, x2}={4, -4}
STEP
6
:
Check that the first solution is correct :
Original equation, root isolated, after tidy up
√5-x = -√5+x+4
Plug in 4 for x
√5-(4) = -√5+(4)+4
Simplify
√1 = 1
Solution checks !!
Solution is:
x = 4
STEP
7
:
Check that the second solution is correct :
Original equation, root isolated, after tidy up
√5-x = -√5+x+4
Plug in -4 for x
√5-(-4) = -√5+(-4)+4
Simplify
√9 = 3
Solution checks !!
Solution is:
x = -4
Two solutions were found :
x = -4
x = 4