Tg*x(3*tg*x-4)=2*tg*x-3

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вопрос

To solve this equation, we first distribute the tg*x to both terms on the left side of the equation:

tgx 3 tgx - tgx 4 = 2 tgx - 3

This simplifies to:

3(tg^2(x)) - 4(tg(x)) = 2(tg(x)) - 3

Next, let's substitute u = tg(x) to simplify the equation:

3u^2 - 4u = 2u - 3

Now, let's move all terms to one side of the equation to set it equal to zero:

3u^2 - 6u - 3 = 0

Dividing by 3 to simplify:

u^2 - 2u - 1 = 0

Now, we can solve this quadratic equation using the quadratic formula:

u = (2 ± √(2^2 - 4 1 -1)) / 2

u = (2 ± √(4 + 4)) / 2

u = (2 ± √8) / 2

u = (2 ± 2√2) / 2

Two possible solutions for u are:

u1 = (2 + 2√2) / 2 = 1 + √2
u2 = (2 - 2√2) / 2 = 1 - √2

Finally, substitute back u = tg(x):

tg(x) = 1 ± √2

Therefore, the solutions for x are:

x = arctan(1 + √2)
x = arctan(1 - √2)