Equation roots .Find the roots of the equation (4^2)^x-3*4^x+2=0 .

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the roots of (4^2)^x-3*4^x+2=0

(4^2)^x-3*4^x+2=0

Let 4^x = y

=> y^2 - 3y + 2 = 0

=> y^2 - 2y - y + 2 = 0

=> y(y - 2)  - 1(y - 2) = 0

=> (y - 1)(y - 2) = 0

=> y = 1 and y = 2

4^x = 1 and 4^x = 2

=> x = 0 and x = 1/2

The required roots are x = 0 and x = 1/2

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll substitute 4^x by the variable, t.

t^2 - 3t + 2 = 0

We'll apply quadratic formula to determine t:

t1 = [3+sqrt (9-4*2)]/2*1

t1 = [3+sqrt (1)]/2

t1 = (3+1)/2

t1 = 2

t2 = (3-1)/2

t2 = 1

We'll find the values of x, now.

4^x = t1

4^x=2

We'll write 4^x = 2^2x

2^2x = 2^1

Since the bases are matching, we'll apply one to one property:

2x = 1

x = 1/2

4^x = t2

4^x = 1

4^x = 4^0

x = 0

The solutions of the equation are {0 ; 1/2}.

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