Question

Resolver las siguientes ecuaciones de segundo grado utilizando la fórmula general:

1.
2 − 5 + 6 = 0

2. 2
2 − 7 + 3 = 0

3. −
2 + 7 − 10 = 0

4.
2 − 2 + 1 = 0

5.
2 + + 1 = 0

6.
2 − 4 + 4 = 0

7.
2 − 5 + 6 = 0

8. 7
2 + 21 − 28 = 0

9.
2 + (7 − )
2 = 25
10.
2 + 3 + 2 = 0​

Answer 1

1. x² - 5x + 6 = 0

a = 1

b = -5

c = 6

x = -b ± √b² - 4ac / 2a

x = -(-5) ± √(-5)² - 4 (1) (6) / 2

x = 5 ± √25 - 24/2

x = 5 ± √1/2

x = 5 ± 1/2

x1 = 5 + 1/2 = 3

x2 = 5 - 2/3 = 1

2. 2x² - 7x + 3 = 0

a =2

b = -7

c = 3

x = -b ± √b² - 4ac / 2a

x = -(-7) ± √(-7)² - 4 (2) (3) / 2(2)

x = 7 ± √49 - 24/ 4

x = 7 ± √25/4

x = 7 ± 5/4

x1 = 7 + 5/4 = 3

x2 = 7 - 5/4 = 1/2

Verificación

2x² - 7x + 3 = 0

2(3)² - 7(3) + 3 = 0

2 (9) - 21 + 3 = 0

18 - 21 + 3 = 0

21 - 21 = 0

0 = 0

2x² - 7x + 3 = 0

2(1/2)² - 7(1/2) + 3 = 0

2(1/4) - 7/2 + 3 = 0

1/2 - 7/2 + 3 = 0

-6/2 + 3 = 0

-3 + 3 = 0

0 = 0

3. -x² + 7x - 10 = 0

a = -1

b = 7

c = -10

x = -b ± √b² - 4ac / 2a

x = -7 ± √7² - 4 (-1) (-10) / 2(-1)

x = -7 ± √49 - 40/ -2

x = -7 ± √9/ -2

x = -7 ± 3/-2

x1 = -7 + 3/-2 = 2

x2 = -7 - 3/-2 = 5

4. x² - 2x + 1 = 0

a = 1

b = -2

c = 1

x = -b ± √b² - 4ac / 2a

x = -(-2) ± √(-2)² - 4 (1) (1) / 2(1)

x = 2 ± √4 - 4/2

x = 2 ± 0/2

x1 = 2 + 0/2 = 1

x2 = 2 - 0/2 = 1

Verificación

x² - 2x + 1 = 0

(1)² - 2(1) + 1 = 0

1 - 2 + 1 = 0

2 - 2 = 0

0 = 0

5. x² + x + 1 = 0

a = 1

b = 1

c = 1

x = -b ± √b² - 4ac / 2a

x = -1 ± √1² - 4 (1) (1) / 2

x = -1 ± √1 - 4/2

x = -1 + √-3/2

x1 = -1 + √-3/2

x2 = -1 - √-3/2

6. x² - 4x + 4 = 0

a = 1

b = -4

c = 4

x = -b ± √b² - 4ac / 2a

x = -(-4) ± √(-4)² - 4 (1) (4) / 2(1)

x = 4 ± √16 - 16/2

x = 4 ± 0/2

x1 = 4 + 0/2 = 2

x2 = 4 - 0/2 = 2

Verificación

x² - 4x + 4 = 0

(2)² - 4(2) + 4 = 0

4 - 8 + 4 = 0

8 - 8 = 0

0 = 0

7. x² - 5x + 6 = 0

a = 1

b = -5

c = 6

x = -b ± √b² - 4ac / 2a

x = -(-5) ± √(-5)² - 4 (1) (6) /2(1)

x = 5 ± √25 - 24/2

x = 5 ± √1/2

x = 5 ± 1/2

x1 = 5 + 1/2 = 3

x2 = 5 - 1/2 = 2

8. 7x² + 21x - 28 = 0

a = 7

b = 21

c = -28

x = -b ± √b² - 4ac / 2a

x = -21 + √21² - 4 (7) (-28) / 2(7)

x = -21 ± √441 - (-784) / 14

x = -21 ± √441 + 784 /14

x = -21 ± √1225/14

x = -21 ± 35/14

x1 = -21 + 35/14 = 1

x2 = -21 - 35/14 = -4

Verificación

7x² + 21x - 28 = 0

7(1)² + 21(1) - 28 = 0

7 + 21 - 28 = 0

28 - 28 = 0

0 = 0

7x² + 21x - 28 = 0

7(-4)² + 21(-4) - 28 = 0

7 (16) - 84 - 28 = 0

112 - 112 = 0

0 = 0

9. x² + (7 - x)² = 25

x² + (7² - 2 (x) (7) + x²) = 25

x² + (49 - 14x + x²) - 25 = 0

x² + x² - 14x + 49 - 25 = 0

2x² - 14x + 24 = 0

a = 2

b = -14

c = 24

x = -b ± √b² - 4ac / 2a

x = -(-14) ± √(-14)² - 4 (2) (24) / 2(2)

x = 14 ± √196 - 192 / 4

x = 14 ± √4/4

x = 14 ± 2/4

x1 = 14 + 2/4 = 4

x2 = 14 - 2/4 = 3

10. x² + 3x + 2 = 0

a = 1

b = 3

c = 2

x = -b ± √b² - 4ac / 2a

x = -3 ± √(3)² - 4 (1) (2) / 2(1)

x = -3 ± √9 - 8/2

x = -3 ± √1/2

x = -3 ± 1/2

x1 = -3 + 1/2 = -1

x2 = -3 - 1/2 = -2

Verificación

x² + 3x + 2 = 0

(-1)² + 3(-1) + 2 = 0

1 - 3 + 2 = 0

-3 + 3 = 0

0 = 0

x² + 3x + 2 = 0

(-2)² + 3(-2) + 2 = 0

4 - 6 + 2 = 0

6 - 6 = 0

0 = 0

Espero haberte ayudado. Suerte en los estudios ≧◠◡◠≦.