Question

(x^2 - 6x)^2 - 2(x-3)^2=81(Решите уравнение)

Answer 1

$(x^{2} - 6x)^{2} - 2(x - 3)^{2} = 81$

$(x^{2} - 6x + 9 - 9)^{2} - 2(x - 3)^{2} - 81 = 0$

$((x - 3)^{2} - 9)^{2} - 2(x - 3)^{2} - 81 = 0$

$(x - 3)^{4} - 18(x - 3)^{2} + 81 - 2(x - 3)^{2} - 81 = 0$

$(x - 3)^{4} - 20(x - 3)^{2} = 0$

$(x - 3)^{2}((x - 3)^{2} - 20) = 0$

$(x - 3)^{2}(x^{2} - 6x + 9 - 20) = 0$

$(x - 3)^{2}(x^{2} - 6x - 11) = 0$

$1) \ (x - 3)^{2} = 0; \ \ \ x - 3 = 0; \ \ \ x = 3$

$2) \ x^{2} - 6x - 11 = 0$

$D = (-6)^{2} - 4 \cdot 1 \cdot (-11) = 36 + 44 = 80$

$x_{1,2} = \dfrac{6 \pm 4\sqrt{5}}{2} = \dfrac{2(3 \pm 2\sqrt{5})}{2} = 3 \pm 2\sqrt{5} = \left[\begin{array}{ccc}x_{1} = 3 - 2\sqrt{5}\\x_{2} = 3 + 2\sqrt{5}\\\end{array}\right$

Ответ: $x = 3; \ x = 3 - 2\sqrt{5}; \ x = 3 +2\sqrt{5}$