Вправи: 855

855. Розв’яжіть рівняння, використовуючи формулу коренів, та перевірте істинність теореми Вієта для кожного з рівнянь:

1) x² + 4x – 5 = 0;

D = 4² – 4 • 1 • (–5) = 36; D = 6;

х₁ = -4+62 = 1; х₂ = -4-62 = –5.

х₁ + х₂ = 1 + (–5) = –4 = –p;

х₁х₂ = 1 • (–5) = –5 = q;

2) x² – 4x – 21 = 0;

D = (–4)² – 4 • 1 • (–21) = 100; D = 10;

х₁ = 4+102 = 7; х₂ = 4-102 = –3.

х₁ + х₂ = 7 + (–3) = 4 = –(–4) = –p;

х₁х₂ = 7 • (–3) = –21 = q.

3) 2x² – 5x + 3 = 0.

D = (–5)² – 4 • 2 • 3 = 1; D = 1;

х₁ = 5+12 • 2 = 1,5; х₂ = 5-12 • 2 = 1.

х₁ + х₂ = 1,5 + 1 = 2,5; –ba = –-52 = 2,5; х₁ + х₂ = –ba;

х₁х₂ = 1,5 • 1 = 1,5; ca = 32 = 1,5; х₁х₂ = ca.

4) 2x² + 5x + 2 = 0;

D = 5² – 4 • 2 • 2 = 9; D = 3;

х₁ = -5+32 • 2 = -24 = –0,5; х₂ = -5-34 = –2.

х₁ + х₂ = –0,5 + (–2) = –2,5; –ba = –52 = –2,5; х₁ + х₂ = –ba;

х₁х₂ = –0,5 • (–2) = 1; ca = 22 = 1; х₁х₂ = ca.