Practica + - page 20

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Ecuaciones de segundo grado
Ecuaciones
. Resuelve las siguientes ecuaciones de segundo grado.
a) x2 − 5 x + 6 = 0 → a = 1 b = −5 c = 6
b) x2 − x − 2 = 0 → a = 1 b = − 1 c = −2
c) x2 + 6 x + 9 = 0 → a = 1 b = 6 c = 9
d) 4 x 2 − 13 x + 3 = 0 → a = 4 b = −13 c = 3
e) x2 − x − 1 = 0 → a = 1 b = −1 c = −1
f) − x2 − 5 x + 6 = 0 → a = −1 b = −5 c = 6
. Resuelve estas ecuaciones de segundo grado incompletas.
a) x2 − 5 x = 0 c) 6 x 2 − x = 0
b) x2 − 49 = 0 d) 4 x2 − 9 = 0
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 = −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 = 1 ± ( −1)2 − 4 ⋅ 1⋅ ( −1)
2 ⋅ 1
= 1± 1+ 4
2
= 1 ± 5
2
→
x = 1 + 5
2
x = 1− 5
2
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪
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
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x (6 x − 1) = 0 →
x = 0
6 x − 1 = 0 → x = 1
6
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪⎪
, b , c = 6
, b , c = −2
, b = , c = 9
, , c
, b , c = −1
, , c = 6
x ( x − 5) = 0 → x = 0
x − 5 = 0 → x = 5
⎧
⎨
⎪⎪
⎩⎪⎪
x2 = 49 x = ± 49 x = 7
x = 7
4 x 2 = 9 x 2 = 9
4
x = ± 9
4
x = 3
2
x = 3
2