Страница 102 номер 392, ГДЗ по алгебре за 9 класс к учебнику Мерзляка

1) \(x^{4} - 13x^{2} + 36 = 0\)

Замена \(x^{2} = t\)

\(t^{2} - 13t + 36 = 0\)
\(D = 13^{2} - 4 \cdot 36 = 169 - 144 = 25 = 5^{2}\)
\(t_{1} = \dfrac{13 - 5}{2} = \dfrac{8}{2} = 4 \Rightarrow x_{1{,}2} = \pm 2\)
\(t_{2} = \dfrac{13 + 5}{2} = \dfrac{18}{2} = 9 \Rightarrow x_{3{,}4} = \pm 3\)

2) \(x^{4} - 5x^{2} - 6 = 0\)

Замена \(x^{2} = t\)

\(t^{2} - 5t - 6 = 0\)
\(D = 5^{2} + 4 \cdot 6 = 25 + 24 = 49 = 7^{2}\)
\(t_{1} = \dfrac{5 - 7}{2} = - \dfrac{2}{2} = - 1\)
\(t_{2} = \dfrac{5 + 7}{2} = 6 \Rightarrow x_{1{,}2} = \pm \sqrt{6}\)

3) \(x^{4} + 9x^{2} + 8 = 0\)

Замена \(x^{2} = t\)

\(t^{2} + 9t + 8 = 0\)
\(D = 9^{2} - 4 \cdot 8 = 81 - 32 = 49 = 7^{2}\)
\(t_{1} = \dfrac{- 9 - 7}{2} = - \dfrac{16}{2} = - 8 \Rightarrow x^{2} = - 8\)
\(t_{2} = \dfrac{- 9 + 7}{2} = - \dfrac{2}{2} = - 1 \Rightarrow x^{2} = - 1\)

У данного уравнения нет решения.

4) \(x^{4} - 16x^{2} = 0\)

Замена \(x^{2} = t\)

\(t^{2} - 16t = 0\)
\(t(t - 16) = 0\)
\(t_{1} = 0 \Rightarrow x_{1} = 0\)
\(t_{2} = 16 \Rightarrow x_{2{,}3} = \pm 4\)

Ответ: 1) \(x_{1{,}2} = \pm 2;\ x_{3{,}4} = \pm 3\); 2) \(x_{1{,}2} = \pm \sqrt{6}\); 3) Нет решения; 4) \(x_{1} = 0;\ x_{2{,}3} = \pm 4\).