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Ecuaciones
Ecuaciones de segundo grado
. Resuelve las siguientes ecuaciones de segundo grado incompletas, aplicando la fórmula.
a) x2 + 15 x = 0
b) x2 − 64 = 0
c) x2 − 5 x = 0
d) 6 x2 − x = 0
e) x2 − 49 = 0
. Resuelve nuevamente las ecuaciones del ejercicio anterior, ahora sin utilizar la fórmula. ¿Qué
método te parece mejor? ¿Por qué?
a) d)
b) e)
c)
x = −15 ± 152 − 4 ⋅ 1⋅ 0
2 ⋅ 1
= −15 ± 225 − 0
2
= −15 ± 225
2
= −15 ± 15
2
→
x = −15 + 15
2
→ x = 0
x = −15 − 15
2
→ x = −15
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 0 ± 02 − 4 ⋅ 1⋅ (−64 )
2 ⋅ 1
= 0 ± 0 + 256
2
= 0 ± 256
2
= 0 ± 16
2
→
x = 16
2
→ x = 8
x = −16
2
→ x = −8
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 5 ± ( −5)2 − 4 ⋅1⋅ 0
2 ⋅ 1
= 5 ± 25 − 0
2
= 5 ± 25
2
= 5 ± 5
2
→
x = 5 + 5
2
→ x = 5
x = 5 − 5
2
→ x = 0
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 1 ± ( −1)2 − 4 ⋅ 6 ⋅ 0
2 ⋅ 6
= 1 ± 1− 0
12
= 1± 1
12
= 1 ± 1
12
→
x = 1 + 1
12
= 2
12
→ x = 1
6
x = 1 − 1
12
→ x = 0
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
x = 0 ± 02 − 4 ⋅ 1⋅ ( −49)
2 ⋅ 1
= 0 ± 0 + 196
2
= 0 ± 196
2
= 0 ± 14
2
→
x = 14
2
→ x = 7
x = −14
2
→ x = −7
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
6 x 2 − x = 0
x (6 x − 1) = 0 →
x = 0
6 x − 1 = 0 → x = 1
6
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
a = 1, b = 15, c = 0
a = 1, b = 0, c = − 64
a = 1, b = − 5, c = 0
a = 6, b = − 1, c = 0
a = 1, b = 0, c = − 49
x2 − 49 = 0
x2 = 49 → x = ± 49 → x = 7
x = −7
⎧
⎨
⎪⎪
⎩⎪⎪
x2 + 15 x = 0
x ( x + 15) = 0 → x = 0
x + 15 = 0 → x = −15
⎧
⎨
⎪⎪
⎩⎪⎪
x2 − 64 = 0
x2 = 64 → x = ± 64 → x = 8
x = −8
⎧
⎨
⎪⎪
⎩⎪⎪
Es mejor resolver las ecuaciones incompletas sin la fórmula, porque es más rápido y más sencillo.
x2 − 5 x = 0
x ( x − 5) = 0 → x = 0
x − 5 = 0 → x = 5
⎧
⎨
⎪⎪
⎩⎪⎪