Log((x^2-4*x+3)/4,sqrt(2))=-2

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вопрос

To solve this logarithmic equation, we need to rewrite the given expression in a simpler form.

First, let's simplify the expression inside the logarithm:
(x^2 - 4x + 3)/4 = ((x-1)(x-3))/4

Now, the given equation becomes:
log(((x-1)(x-3))/4, sqrt(2)) = -2

Since the base of the logarithm is sqrt(2), we can rewrite the equation as:
((x-1)(x-3))/4 = sqrt(2)^-2
((x-1)(x-3))/4 = 1/2

Multiplying both sides by 4:
(x-1)(x-3) = 2

Expanding the left side:
x^2 - 4x + 3 = 2
x^2 - 4x + 1 = 0

This is a quadratic equation that can be solved using the quadratic formula:
x = (4 ± sqrt((-4)^2 - 411)) / (2*1)
x = (4 ± sqrt(16 - 4)) / 2
x = (4 ± sqrt(12)) / 2
x = (4 ± 2sqrt(3)) / 2

Therefore, the solutions to the equation are:
x = 2 ± sqrt(3)