Тренажер «Квадратные уравнения» Реши уравнения: 1. -2x²+5x+3=0 2. x²-22x-23=0 3. x²-2x+5=0 4. x²+6x+8=0 5. x²-34x+289=0 6. 5x²-8x+3=0 7. 3x²-8x+5=0 8. 5x²+26x-24=0 9. x2=4x+96 10. 25=26x-x² 11. x²-5x+3=0 12. x²+6x+3=0 13. x²-12x=0 14. -x²+7x=0 15. x²-49=0 16. -5x2+9=0 17. 81x²=0 18. 3x²-75=0 19. 8x(1+2x)= -1 20. 5(x+2)²-6x-44 Вариант 3

Решим квадратные уравнения Варианта 3.

1. -2x² + 5x + 3 = 0

   2x² - 5x - 3 = 0

   D = (-5)² - 4 × 2 × (-3) = 25 + 24 = 49

   x₁ = (5 + √49) / (2 × 2) = (5 + 7) / 4 = 12 / 4 = 3

   x₂ = (5 - √49) / (2 × 2) = (5 - 7) / 4 = -2 / 4 = -1/2

   Ответ: x₁ = 3, x₂ = -1/2

2. x² - 22x - 23 = 0

   D = (-22)² - 4 × 1 × (-23) = 484 + 92 = 576

   x₁ = (22 + √576) / (2 × 1) = (22 + 24) / 2 = 46 / 2 = 23

   x₂ = (22 - √576) / (2 × 1) = (22 - 24) / 2 = -2 / 2 = -1

   Ответ: x₁ = 23, x₂ = -1

3. x² - 2x + 5 = 0

   D = (-2)² - 4 × 1 × 5 = 4 - 20 = -16

   Т.к. D < 0, то уравнение не имеет действительных корней.

   Ответ: нет действительных корней

4. x² + 6x + 8 = 0

   D = 6² - 4 × 1 × 8 = 36 - 32 = 4

   x₁ = (-6 + √4) / (2 × 1) = (-6 + 2) / 2 = -4 / 2 = -2

   x₂ = (-6 - √4) / (2 × 1) = (-6 - 2) / 2 = -8 / 2 = -4

   Ответ: x₁ = -2, x₂ = -4

5. x² - 34x + 289 = 0

   D = (-34)² - 4 × 1 × 289 = 1156 - 1156 = 0

   x = 34 / (2 × 1) = 17

   Ответ: x = 17

6. 5x² - 8x + 3 = 0

   D = (-8)² - 4 × 5 × 3 = 64 - 60 = 4

   x₁ = (8 + √4) / (2 × 5) = (8 + 2) / 10 = 10 / 10 = 1

   x₂ = (8 - √4) / (2 × 5) = (8 - 2) / 10 = 6 / 10 = 3 / 5

   Ответ: x₁ = 1, x₂ = 3/5

7. 3x² - 8x + 5 = 0

   D = (-8)² - 4 × 3 × 5 = 64 - 60 = 4

   x₁ = (8 + √4) / (2 × 3) = (8 + 2) / 6 = 10 / 6 = 5 / 3

   x₂ = (8 - √4) / (2 × 3) = (8 - 2) / 6 = 6 / 6 = 1

   Ответ: x₁ = 5/3, x₂ = 1

8. 5x² + 26x - 24 = 0

   D = 26² - 4 × 5 × (-24) = 676 + 480 = 1156

   x₁ = (-26 + √1156) / (2 × 5) = (-26 + 34) / 10 = 8 / 10 = 4 / 5

   x₂ = (-26 - √1156) / (2 × 5) = (-26 - 34) / 10 = -60 / 10 = -6

   Ответ: x₁ = 4/5, x₂ = -6

9. x² = 4x + 96

   x² - 4x - 96 = 0

   D = (-4)² - 4 × 1 × (-96) = 16 + 384 = 400

   x₁ = (4 + √400) / (2 × 1) = (4 + 20) / 2 = 24 / 2 = 12

   x₂ = (4 - √400) / (2 × 1) = (4 - 20) / 2 = -16 / 2 = -8

   Ответ: x₁ = 12, x₂ = -8

10. 25 = 26x - x²

    x² - 26x + 25 = 0

    D = (-26)² - 4 × 1 × 25 = 676 - 100 = 576

    x₁ = (26 + √576) / (2 × 1) = (26 + 24) / 2 = 50 / 2 = 25

    x₂ = (26 - √576) / (2 × 1) = (26 - 24) / 2 = 2 / 2 = 1

    Ответ: x₁ = 25, x₂ = 1

11. x² - 5x + 3 = 0

    D = (-5)² - 4 × 1 × 3 = 25 - 12 = 13

    x₁ = (5 + √13) / (2 × 1) = (5 + √13) / 2

    x₂ = (5 - √13) / (2 × 1) = (5 - √13) / 2

    Ответ: x₁ = (5 + √13) / 2, x₂ = (5 - √13) / 2

12. x² + 6x + 3 = 0

    D = 6² - 4 × 1 × 3 = 36 - 12 = 24

    x₁ = (-6 + √24) / (2 × 1) = (-6 + 2√6) / 2 = -3 + √6

    x₂ = (-6 - √24) / (2 × 1) = (-6 - 2√6) / 2 = -3 - √6

    Ответ: x₁ = -3 + √6, x₂ = -3 - √6

13. x² - 12x = 0

    x(x - 12) = 0

    x₁ = 0

    x - 12 = 0

    x = 12

    x₂ = 12

    Ответ: x₁ = 0, x₂ = 12

14. -x² + 7x = 0

    x(-x + 7) = 0

    x₁ = 0

    -x + 7 = 0

    x = 7

    x₂ = 7

    Ответ: x₁ = 0, x₂ = 7

15. x² - 49 = 0

    x² = 49

    x₁ = √49 = 7

    x₂ = -√49 = -7

    Ответ: x₁ = 7, x₂ = -7

16. -5x² + 9 = 0

    5x² = 9

    x² = 9 / 5

    x₁ = √(9 / 5) = 3 / √5 = (3√5) / 5

    x₂ = -√(9 / 5) = -3 / √5 = (-3√5) / 5

    Ответ: x₁ = (3√5) / 5, x₂ = (-3√5) / 5

17. 81x² = 0

    x² = 0

    x = 0

    Ответ: x = 0

18. 3x² - 75 = 0

    3x² = 75

    x² = 25

    x₁ = √25 = 5

    x₂ = -√25 = -5

    Ответ: x₁ = 5, x₂ = -5

19. 8x(1 + 2x) = -1

    8x + 16x² = -1

    16x² + 8x + 1 = 0

    D = 8² - 4 × 16 × 1 = 64 - 64 = 0

    x = -8 / (2 × 16) = -8 / 32 = -1 / 4

    Ответ: x = -1/4

20. 5(x + 2)² - 6x - 44

    5(x² + 4x + 4) - 6x - 44 = 0

    5x² + 20x + 20 - 6x - 44 = 0

    5x² + 14x - 24 = 0

    D = 14² - 4 × 5 × (-24) = 196 + 480 = 676

    x₁ = (-14 + √676) / (2 × 5) = (-14 + 26) / 10 = 12 / 10 = 6 / 5

    x₂ = (-14 - √676) / (2 × 5) = (-14 - 26) / 10 = -40 / 10 = -4

    Ответ: x₁ = 6/5, x₂ = -4