Practica + - page 36

1060 y 1081
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. Resuelve estas ecuaciones de segundo grado.
a) x2 − x − 1 = 0 → a = 1 b = −1 c = −1
b) 2 x 2 + 3 x − 5 = 0
c) 7 x 2 + 5 x + 2 = 0
d) 49 x 2 − 56 x + 16 = 0
e) 40 x2 + 69 x 8 = 0
f) x2 − 21x + 110 = 0
g) x2 − 7 x + 12 = 0
h) 9 x 2 + 80 x − 9 = 0
Ecuaciones de segundo grado
Ecuaciones
x = 1 ± (−1)2 − 4 ⋅ 1⋅ (−1)
2 ⋅1
= 1 ± 1 + 4
2
= 1± 5
2
→
x = 1 + 5
2
x = 1− 5
2
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪
x = −3 ± 32 − 4 ⋅ 2 ⋅ (−5)
2 ⋅ 2
= −3 ± 9 + 40
4
= −3 ± 49
4
= −3 ± 7
4
→
x = −3 + 7
4
→ x = 1
x = −3 − 7
4
= − 10
4
→ x = − 5
2
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 56 ± (−56)2 − 4 ⋅ 49 ⋅16
2 ⋅ 49
= 56 ± 3 136 − 3 136
98
= 56 ± 0
98
= 56 ± 0
98
= 56
98
→ x = 4
7
(solución doble)
x = 69 ± (− 69)2 − 4 ⋅ 40 ⋅ 8
2 ⋅ 40
= 69 ± 4761 − 1280
80
= 69 ± 3481
80
= 69 ± 59
80
→
x = 69 + 59
80
= 128
80
→ x = 8
5
x = 69 − 59
80
= 10
80
→ x = 1
8
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 21± ( −21)2 − 4 ⋅1 ⋅110
2 ⋅ 1
= 21 ± 441 − 440
2
= 21± 1
2
= 21 ± 1
2
→
x = 21+ 1
2
→ x = 11
x = 21− 1
2
→ x = 10
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 7 ± (−7)2 − 4 ⋅ 1⋅ 12
2 ⋅ 1
= 7 ± 49 − 48
2
= 7 ± 1
2
= 7 ± 1
2
→
x = 7 + 1
2
→ x = 4
x = 7 − 1
2
→ x = 3
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = −80 ± 802 − 4 ⋅ 9 ⋅ ( −9)
2 ⋅ 9
= −80 ± 6 400 + 324
18
= −80 ± 6724
18
= −80 ± 82
18
→
x = −80 + 82
18
= 2
18
→ x = 1
9
x = −80 − 82
18
→ x = −9
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
, , c = −1
a = 2, b = 3, c = −5
a = 49, b = −56, c = 16
a = 40, b = −69, c = 8
a = 1, b = −21, c = 110
a = 1, b = −7, c = 12
a = 9, b = 80, c = − 9
a = 7, b = 5, c = 2
x = −5 ± 52 − 4 ⋅ 7 ⋅ 2
2 ⋅ 7
= −5 ± 25 − 56
14
= −5 ± − 31
14
No tiene solución real, ya que aparece la raíz cuadrada de un número negativo.