Відповіді на вправи: 515 (1-2)

1) x + y = 9;

x² + y² = 65;

y = 9 – x;

x² + (9 – x)² = 65;

x² + 81 – 18x + x² –65 = 0;

2x² – 18x + 16 = 0;

D = (–18)² – 4 • 2 • 16 = 186 = 14²;

x₁ = $\frac{18-14}{4}$ = $\frac{4}{4}$ = 1;

x₂ = $\frac{18+14}{4}$ = $\frac{32}{4}$ = 8;

2) x – y = 3;

x² – y² = 29;

y = x – 3;

x² + (x – 3)² – 29 = 0;

x² + x² – 6x + 9 – 29 = 0;

2x² – 6x – 20 = 0 |: 2;

x² – 3x– 10 = 0;

x₁ = –2, x₂ = 5;

x = –2; x = 4

y = –2 – 3; y = 4 – 7;

x = –2; x = 5.

y = –5; y = 2.

x = 1

y = 9 – 1

x = 8;

y = 9 – 8;

x = –2

Відповідь: (–2; –5), (5; 2).

x = 1

y = 8

x = 8;

y = 1.

Відповідь: (1; 8), (8; 1).

3) x – 3y = 1;

x² = 5 – y²;

4) x + y = 5;

x² – 1 = y² – 4xy;

x = 1 + 3y;

(1 + 3y)² – 5 + y² = 0;

1 + 6y + 9y² – 5 + y² = 0;

10y² + 6y – 4 = 0| : 2;

5y² + 3y – 2 = 0;

D = 3² – 4 • 5 • (–2) = 49 = 7²;

y₁ = $\frac{-3-7}{10}$ = $\frac{-10}{10}$ = –1;

y₂ = $\frac{-3+7}{10}$ = $\frac{4}{10}$ = $\frac{2}{5}$;

x = –1;

y = 1 + 3 • (–1);

x = $\frac{2}{5}$;

y = 1 + 3 • $\frac{2}{5}$;

y = 5 – x;

x² – 1 – (5 – x)² + 4x(5 – x) = 0;

x² – 1 –(25 – 10x + x²) + 20x – 4x² = 0;

x² – 1 – 25 + 10x – x² + 20x – 4x² = 0;

–4x² + 30x – 26 = 0|: 2;

–2x² + 15x – 13 = 0;

D = 15² – 4 • (–2) • (–13) = 221 = 11²;

x₁ = $\frac{-15-11}{-4}$ = $\frac{-26}{-4}$ = $\frac{13}{2}$ = 6,5;

x₂ = $\frac{-15+11}{-4}$ = $\frac{-4}{-4}$ = 1.

Відповідь: (–2; –1), (2$\frac{1}{5}$; $\frac{2}{5}$).

x = 6,5;

y = 5 – 6,5;

x = 1;

y = 5 – 1;

x = 6,5;

y = –1,5;

x = 1;

y = 4.

Відповідь: (6,5; –1,5), (1; 5).