Вправи: 863 (1)

863. Розв’яжіть рівняння:

1) x⁴ – (n² + 9)x² + 9n² = 0;

t = x² (t ≥ 0);

t² – (n² + 9) t + 9n² = 0;

D = (–(n² + 9))² – 4 • 1 • 9n² = n⁴ + 18n² + 81 – 36n² = n⁴ – 18n² + 81 = (n² – 9)².

t₁ = n2+ 9-(n2- 9)22 •1 = n2+ 9- (n2- 9)2 = n2+ 9- n2+ 92 = 182 = 9.

t₂ = n2+ 9+(n2- 9)22 •1 = n2+ 9+ n2- 92 •1 = 2n22 = n².

x² = 9 або x² = n²;

x = ±3. x = ± n2;

x = ± n.

Відповідь: ±3; ± n.

2) x + 1n = n + 1x; x^\x – 1x + 1n – n^\x = 0;

x2- 1+1n- nxx = 0; ОДЗ: x ≠ 0.

x² – 1 + (1n – n)x = 0;

x² + (1n – n)x – 1 = 0;

D = (1n – n)² – 4 • 1 • (–1) = 1n2 – 2 • 1n • n + n² + 4 = 1n2 + n2\n2 + 2\n2 = 1+ n4+ 2n2n2 = (1+n2)2n2 = (1+n2n)².

x₁ = -1n- n- (1+ n2n)22 •1 = -1n+ n\n- 1+n2n2 = 1+ n2-1- n2n2 = -22n = –1n.

x₂ = -1n- n+ (1+ n2n)22 •1 = -1n+ n\n+ 1+n2n2 = 1+ n2+1+ n2n2 = 2n22n = n.

Відповідь: n; –1n.