Practica + - page 35

1060 y 1081
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Ecuaciones
Ecuaciones de segundo grado
. Resuelve las siguientes ecuaciones de segundo grado.
a) x2 − 5 x + 6 = 0 → a = 1 b = −5 c = 6
b) x2 − 12x + 27 = 0
c) x2 − x − 2 = 0 → a = 1 b = − 1 c = −2
d) x2 + 16 x + 60 = 0
e) x2 + 6 x + 9 = 0 → a = 1 b = 6 c = 9
f) 4 x 2 − 13 x + 3 = 0 → a = 4 b = −13 c = 3
g) 15 x 2 + 7 x − 2 = 0
h) − x2 − 5 x + 6 = 0 → a = −1 b = −5 c = 6
x = 5 ± (−5)2 − 4 ⋅ 1⋅ 6
2 ⋅1
= 5 ± 25 − 24
2
= 5 ± 1
2
= 5 ± 1
2
→
x = 5 + 1
2
→ x = 3
x = 5 − 1
2
→ x = 2
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 1± (−1)2 − 4 ⋅ 1⋅ (−2)
2 ⋅ 1
= 1 ± 1 + 8
2
= 1 ± 9
2
= 1 ± 3
2
→
x = 1 + 3
2
→ x = 2
x = 1− 3
2
→ x = −1
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = −16 ± 162 − 4 ⋅1⋅ 60
2 ⋅ 1
= −16 ± 256 − 240
2
= −16 ± 16
2
= −16 ± 4
2
→
x = −16 + 4
2
→ x = −6
x = −16 − 4
2
→ x = −10
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = −6 ± 62 − 4 ⋅1⋅ 9
2 ⋅1
= −6 ± 36 − 36
2
= −6 ± 0
2
= −6 ± 0
2
= −6
2
→ x = −3 (solución doble)
x = 13 ± ( −13)2 − 4 ⋅ 4 ⋅ 3
2 ⋅ 4
= 13 ± 169 − 48
8
= 13 ± 121
8
= 13 ± 11
8
→
x = 13 + 11
8
→ x = 3
x = 13 − 11
8
= 2
8
→ x = 1
4
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = −7 ± 72 − 4 ⋅ 15 ⋅ (−2)
2 ⋅ 15
= −7 ± 49 + 120
30
= −7 ± 169
30
= −7 ± 13
30
→
x = −7 + 13
30
= 6
30
→ x = 1
5
x = −7 −13
30
= −20
30
→ x = − 2
3
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 5 ± ( −5)2 − 4 ⋅ ( −1) ⋅ 6
2 ⋅ (−1)
= 5 ± 25 + 24
−2
= 5 ± 49
−2
= 5 ± 7
−2
→
x = 5 + 7
−2
→ x = −6
x = 5 − 7
−2
→ x = 1
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
, b , c = 6
, b = −1, c = −2
a = 1, b = 16, c = 60
a = 1, b = −12, c = 27
, b = , c = 9
, , c
a = 15, b = 7, c = −2
, , c = 6
x = 12 ± ( −12)2 − 4 ⋅ 1⋅ 27
2 ⋅ 1
= 12 ± 144 − 108
2
= 12 ± 36
2
= 12 ± 6
2
→
x = 12 + 6
2
→ x = 9
x = 12 − 6
2
→ x = 3
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪