The sum of the possible values of x of the equation
4^x - 3 [ 2^(x + 2) ] + 32 = 0 is
(1) 12
(2) 4
(3) 5
(4) 1
Answers
Answer:
Option 3 i.e. 5
Step-by-step explanation:
Given:-
equation 4^x - 3 [ 2^(x + 2) ] + 32 = 0
To find:-
The sum of the possible values of x of the equation 4^x - 3 [ 2^(x + 2) ] + 32 = 0
Solution:-
Given equation is 4^x - 3 [ 2^(x + 2) ] + 32 = 0
=>(2^2)^x - 3[2^(x+2)] +32 = 0
we know that (a^m)^n = a^mn
=>2^2x -3[2^(x+2)] +32 = 0
We know that a^(m+n)=a^m × a^n
=>2^2x - 3[2^x × 2^2] + 32 = 0
=>2^2x -3[2^x ×4] +32 = 0
=>2^2x -12×2^x +32 = 0
=>(2^x)^2 - 12×2^x +32 = 0
Put 2^x = a then it becomes
=>a^2 -12a +32 = 0
=>a^2 -4a -8a +32 = 0
=>a(a-4)-8(a-4) = 0
=>(a - 4)(a - 8) = 0
=>a - 4 = 0 (or) a - 8 = 0
=>a = 4 or a = 8
now
=>2^x = 4 or 2^x = 8
=>2^x = 2^2 or 2^x = 2^3
Since bases are equal then exponents must be equal
=>x = 2 and x = 3
The possible values of x = 2 and 3
The sum of the values of x = 2+3 = 5
Answer:-
The sum of the possible values of x = 5
Used formula:-
- a^m × a^n = a^(m+n)
- (a^m)^n = a^mn