660. Решите уравнение: 1) x² – 3x + 2 = 0; 2) x² + 12x – 13 = 0; 3) x² – 7x + 10 = 0; 4) x² – x – 72 = 0; 5) 2x2 – 5x + 2 = 0; 6) 2x² - 7x - 4 = 0; 7) 4x2 – 3x - 1 = 0; 8) −2x² + x + 15 = 0; 9) 6x² + 7x - 5 = 0; 10) 18x²-9x-5=0; 11) x² – 6x + 11 = 0; 12) -x² - 8x + 12 = 0.

Решение:

Будем решать уравнения через дискриминант и формулу корней квадратного уравнения:

\[ x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} \]

1. x² - 3x + 2 = 0

   D = (-3)² - 4 * 1 * 2 = 9 - 8 = 1

   x₁ = (3 + √1) / 2 = (3 + 1) / 2 = 2

   x₂ = (3 - √1) / 2 = (3 - 1) / 2 = 1

   x₁ = 2, x₂ = 1

2. x² + 12x - 13 = 0

   D = 12² - 4 * 1 * (-13) = 144 + 52 = 196

   x₁ = (-12 + √196) / 2 = (-12 + 14) / 2 = 1

   x₂ = (-12 - √196) / 2 = (-12 - 14) / 2 = -13

   x₁ = 1, x₂ = -13

3. x² - 7x + 10 = 0

   D = (-7)² - 4 * 1 * 10 = 49 - 40 = 9

   x₁ = (7 + √9) / 2 = (7 + 3) / 2 = 5

   x₂ = (7 - √9) / 2 = (7 - 3) / 2 = 2

   x₁ = 5, x₂ = 2

4. x² - x - 72 = 0

   D = (-1)² - 4 * 1 * (-72) = 1 + 288 = 289

   x₁ = (1 + √289) / 2 = (1 + 17) / 2 = 9

   x₂ = (1 - √289) / 2 = (1 - 17) / 2 = -8

   x₁ = 9, x₂ = -8

5. 2x² - 5x + 2 = 0

   D = (-5)² - 4 * 2 * 2 = 25 - 16 = 9

   x₁ = (5 + √9) / 4 = (5 + 3) / 4 = 2

   x₂ = (5 - √9) / 4 = (5 - 3) / 4 = 0,5

   x₁ = 2, x₂ = 0,5

6. 2x² - 7x - 4 = 0

   D = (-7)² - 4 * 2 * (-4) = 49 + 32 = 81

   x₁ = (7 + √81) / 4 = (7 + 9) / 4 = 4

   x₂ = (7 - √81) / 4 = (7 - 9) / 4 = -0,5

   x₁ = 4, x₂ = -0,5

7. 4x² - 3x - 1 = 0

   D = (-3)² - 4 * 4 * (-1) = 9 + 16 = 25

   x₁ = (3 + √25) / 8 = (3 + 5) / 8 = 1

   x₂ = (3 - √25) / 8 = (3 - 5) / 8 = -0,25

   x₁ = 1, x₂ = -0,25

8. -2x² + x + 15 = 0

   D = 1² - 4 * (-2) * 15 = 1 + 120 = 121

   x₁ = (-1 + √121) / (-4) = (-1 + 11) / (-4) = -2,5

   x₂ = (-1 - √121) / (-4) = (-1 - 11) / (-4) = 3

   x₁ = -2,5, x₂ = 3

9. 6x² + 7x - 5 = 0

   D = 7² - 4 * 6 * (-5) = 49 + 120 = 169

   x₁ = (-7 + √169) / 12 = (-7 + 13) / 12 = 0,5

   x₂ = (-7 - √169) / 12 = (-7 - 13) / 12 = -5/3

   x₁ = 0,5, x₂ = -5/3

10. 18x² - 9x - 5 = 0

    D = (-9)² - 4 * 18 * (-5) = 81 + 360 = 441

    x₁ = (9 + √441) / 36 = (9 + 21) / 36 = 5/6

    x₂ = (9 - √441) / 36 = (9 - 21) / 36 = -1/3

    x₁ = 5/6, x₂ = -1/3

11. x² - 6x + 11 = 0

    D = (-6)² - 4 * 1 * 11 = 36 - 44 = -8

    Так как D < 0, уравнение не имеет корней.

12. -x² - 8x + 12 = 0

    D = (-8)² - 4 * (-1) * 12 = 64 + 48 = 112

    x₁ = (8 + √112) / (-2) = (8 + 4√7) / (-2) = -4 - 2√7

    x₂ = (8 - √112) / (-2) = (8 - 4√7) / (-2) = -4 + 2√7

    x₁ = -4 - 2√7, x₂ = -4 + 2√7

Ответ: 1) x₁ = 2, x₂ = 1; 2) x₁ = 1, x₂ = -13; 3) x₁ = 5, x₂ = 2; 4) x₁ = 9, x₂ = -8; 5) x₁ = 2, x₂ = 0,5; 6) x₁ = 4, x₂ = -0,5; 7) x₁ = 1, x₂ = -0,25; 8) x₁ = -2,5, x₂ = 3; 9) x₁ = 0,5, x₂ = -5/3; 10) x₁ = 5/6, x₂ = -1/3; 11) нет корней; 12) x₁ = -4 - 2√7, x₂ = -4 + 2√7.