EJERCICIOS RESUELTOS DE MATEMÁTICAS

ECUACIONES DE SEGUNDO GRADO :

Ejercicio 1
	-3	x2						=	0

a = -3
b = 0
c = 0

x =	________________________   - ( 0 ) ± √ ( 0 )2 - 4 ( -3 )( 0 )   -------------------------------------------------------------   2( -3 )
x =	________________________   0 ± √ 0 + 0   -------------------------------------------------------------   -6
x =	________   0 ± √ 0     ------------------------------------       -6
x =	0 ± 0     -----------------------------     -6

x1 =	0 + 0   --------------     -6		x2 =	0 - 0   --------------     -6
x1 =	0   ----------   -6		x2 =	0   ----------   -6
x1 =	0		x2 =	0

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Ejercicio 2
	-4	x2	+	4	x			=	0

a = -4
b = 4
c = 0

x =	________________________   - ( 4 ) ± √ ( 4 )2 - 4 ( -4 )( 0 )   -------------------------------------------------------------   2( -4 )
x =	________________________   -4 ± √ 16 + 0   -------------------------------------------------------------   -8
x =	________   -4 ± √ 16     ------------------------------------       -8
x =	-4 ± 4     -----------------------------     -8

x1 =	-4 + 4   --------------     -8		x2 =	-4 - 4   --------------     -8
x1 =	0   ----------   -8		x2 =	-8   ----------   -8
x1 =	0		x2 =	1

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Ejercicio 3
	-3	x2	+	3	x			=	0

a = -3
b = 3
c = 0

x =	________________________   - ( 3 ) ± √ ( 3 )2 - 4 ( -3 )( 0 )   -------------------------------------------------------------   2( -3 )
x =	________________________   -3 ± √ 9 + 0   -------------------------------------------------------------   -6
x =	________   -3 ± √ 9     ------------------------------------       -6
x =	-3 ± 3     -----------------------------     -6

x1 =	-3 + 3   --------------     -6		x2 =	-3 - 3   --------------     -6
x1 =	0   ----------   -6		x2 =	-6   ----------   -6
x1 =	0		x2 =	1

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Ejercicio 4
	3	x2	+	3	x			=	0

a = 3
b = 3
c = 0

x =	________________________   - ( 3 ) ± √ ( 3 )2 - 4 ( 3 )( 0 )   -------------------------------------------------------------   2( 3 )
x =	________________________   -3 ± √ 9 + 0   -------------------------------------------------------------   6
x =	________   -3 ± √ 9     ------------------------------------       6
x =	-3 ± 3     -----------------------------     6

x1 =	-3 + 3   --------------     6		x2 =	-3 - 3   --------------     6
x1 =	0   ----------   6		x2 =	-6   ----------   6
x1 =	0		x2 =	-1

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Ejercicio 5
	4	x2	+	4	x			=	0

a = 4
b = 4
c = 0

x =	________________________   - ( 4 ) ± √ ( 4 )2 - 4 ( 4 )( 0 )   -------------------------------------------------------------   2( 4 )
x =	________________________   -4 ± √ 16 + 0   -------------------------------------------------------------   8
x =	________   -4 ± √ 16     ------------------------------------       8
x =	-4 ± 4     -----------------------------     8

x1 =	-4 + 4   --------------     8		x2 =	-4 - 4   --------------     8
x1 =	0   ----------   8		x2 =	-8   ----------   8
x1 =	0		x2 =	-1

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Ejercicio 6
	-4	x2	-	8	x	-	4	=	0

a = -4
b = -8
c = -4

x =	________________________   - ( -8 ) ± √ ( -8 )2 - 4 ( -4 )( -4 )   -------------------------------------------------------------   2( -4 )
x =	________________________   8 ± √ 64 - 64   -------------------------------------------------------------   -8
x =	________   8 ± √ 0     ------------------------------------       -8
x =	8 ± 0     -----------------------------     -8

x1 =	8 + 0   --------------     -8		x2 =	8 - 0   --------------     -8
x1 =	8   ----------   -8		x2 =	8   ----------   -8
x1 =	-1		x2 =	-1

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Ejercicio 7
	-4	x2	+	8	x			=	0

a = -4
b = 8
c = 0

x =	________________________   - ( 8 ) ± √ ( 8 )2 - 4 ( -4 )( 0 )   -------------------------------------------------------------   2( -4 )
x =	________________________   -8 ± √ 64 + 0   -------------------------------------------------------------   -8
x =	________   -8 ± √ 64     ------------------------------------       -8
x =	-8 ± 8     -----------------------------     -8

x1 =	-8 + 8   --------------     -8		x2 =	-8 - 8   --------------     -8
x1 =	0   ----------   -8		x2 =	-16   ----------   -8
x1 =	0		x2 =	2

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Ejercicio 8
	-3	x2	-	6	x			=	0

a = -3
b = -6
c = 0

x =	________________________   - ( -6 ) ± √ ( -6 )2 - 4 ( -3 )( 0 )   -------------------------------------------------------------   2( -3 )
x =	________________________   6 ± √ 36 + 0   -------------------------------------------------------------   -6
x =	________   6 ± √ 36     ------------------------------------       -6
x =	6 ± 6     -----------------------------     -6

x1 =	6 + 6   --------------     -6		x2 =	6 - 6   --------------     -6
x1 =	12   ----------   -6		x2 =	0   ----------   -6
x1 =	-2		x2 =	0

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Ejercicio 9
	-3	x2	+	6	x	-	3	=	0

a = -3
b = 6
c = -3

x =	________________________   - ( 6 ) ± √ ( 6 )2 - 4 ( -3 )( -3 )   -------------------------------------------------------------   2( -3 )
x =	________________________   -6 ± √ 36 - 36   -------------------------------------------------------------   -6
x =	________   -6 ± √ 0     ------------------------------------       -6
x =	-6 ± 0     -----------------------------     -6

x1 =	-6 + 0   --------------     -6		x2 =	-6 - 0   --------------     -6
x1 =	-6   ----------   -6		x2 =	-6   ----------   -6
x1 =	1		x2 =	1

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Ejercicio 10
	-2	x2				+	2	=	0

a = -2
b = 0
c = 2

x =	________________________   - ( 0 ) ± √ ( 0 )2 - 4 ( -2 )( 2 )   -------------------------------------------------------------   2( -2 )
x =	________________________   0 ± √ 0 + 16   -------------------------------------------------------------   -4
x =	________   0 ± √ 16     ------------------------------------       -4
x =	0 ± 4     -----------------------------     -4

x1 =	0 + 4   --------------     -4		x2 =	0 - 4   --------------     -4
x1 =	4   ----------   -4		x2 =	-4   ----------   -4
x1 =	-1		x2 =	1

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Ejercicio 11
	-2	x2	+	4	x			=	0

a = -2
b = 4
c = 0

x =	________________________   - ( 4 ) ± √ ( 4 )2 - 4 ( -2 )( 0 )   -------------------------------------------------------------   2( -2 )
x =	________________________   -4 ± √ 16 + 0   -------------------------------------------------------------   -4
x =	________   -4 ± √ 16     ------------------------------------       -4
x =	-4 ± 4     -----------------------------     -4

x1 =	-4 + 4   --------------     -4		x2 =	-4 - 4   --------------     -4
x1 =	0   ----------   -4		x2 =	-8   ----------   -4
x1 =	0		x2 =	2

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Ejercicio 12
	3	x2	+	6	x			=	0

a = 3
b = 6
c = 0

x =	________________________   - ( 6 ) ± √ ( 6 )2 - 4 ( 3 )( 0 )   -------------------------------------------------------------   2( 3 )
x =	________________________   -6 ± √ 36 + 0   -------------------------------------------------------------   6
x =	________   -6 ± √ 36     ------------------------------------       6
x =	-6 ± 6     -----------------------------     6

x1 =	-6 + 6   --------------     6		x2 =	-6 - 6   --------------     6
x1 =	0   ----------   6		x2 =	-12   ----------   6
x1 =	0		x2 =	-2

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Ejercicio 13
	4	x2				-	4	=	0

a = 4
b = 0
c = -4

x =	________________________   - ( 0 ) ± √ ( 0 )2 - 4 ( 4 )( -4 )   -------------------------------------------------------------   2( 4 )
x =	________________________   0 ± √ 0 + 64   -------------------------------------------------------------   8
x =	________   0 ± √ 64     ------------------------------------       8
x =	0 ± 8     -----------------------------     8

x1 =	0 + 8   --------------     8		x2 =	0 - 8   --------------     8
x1 =	8   ----------   8		x2 =	-8   ----------   8
x1 =	1		x2 =	-1

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Ejercicio 14
	-5	x2	-	15	x	-	10	=	0

a = -5
b = -15
c = -10

x =	________________________   - ( -15 ) ± √ ( -15 )2 - 4 ( -5 )( -10 )   -------------------------------------------------------------   2( -5 )
x =	________________________   15 ± √ 225 - 200   -------------------------------------------------------------   -10
x =	________   15 ± √ 25     ------------------------------------       -10
x =	15 ± 5     -----------------------------     -10

x1 =	15 + 5   --------------     -10		x2 =	15 - 5   --------------     -10
x1 =	20   ----------   -10		x2 =	10   ----------   -10
x1 =	-2		x2 =	-1

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Ejercicio 15
	-5	x2	-	15	x			=	0

a = -5
b = -15
c = 0

x =	________________________   - ( -15 ) ± √ ( -15 )2 - 4 ( -5 )( 0 )   -------------------------------------------------------------   2( -5 )
x =	________________________   15 ± √ 225 + 0   -------------------------------------------------------------   -10
x =	________   15 ± √ 225     ------------------------------------       -10
x =	15 ± 15     -----------------------------     -10

x1 =	15 + 15   --------------     -10		x2 =	15 - 15   --------------     -10
x1 =	30   ----------   -10		x2 =	0   ----------   -10
x1 =	-3		x2 =	0

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Ejercicio 16
	-4	x2	-	12	x	-	8	=	0

a = -4
b = -12
c = -8

x =	________________________   - ( -12 ) ± √ ( -12 )2 - 4 ( -4 )( -8 )   -------------------------------------------------------------   2( -4 )
x =	________________________   12 ± √ 144 - 128   -------------------------------------------------------------   -8
x =	________   12 ± √ 16     ------------------------------------       -8
x =	12 ± 4     -----------------------------     -8

x1 =	12 + 4   --------------     -8		x2 =	12 - 4   --------------     -8
x1 =	16   ----------   -8		x2 =	8   ----------   -8
x1 =	-2		x2 =	-1

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Ejercicio 17
	-4	x2	+	12	x			=	0

a = -4
b = 12
c = 0

x =	________________________   - ( 12 ) ± √ ( 12 )2 - 4 ( -4 )( 0 )   -------------------------------------------------------------   2( -4 )
x =	________________________   -12 ± √ 144 + 0   -------------------------------------------------------------   -8
x =	________   -12 ± √ 144     ------------------------------------       -8
x =	-12 ± 12     -----------------------------     -8

x1 =	-12 + 12   --------------     -8		x2 =	-12 - 12   --------------     -8
x1 =	0   ----------   -8		x2 =	-24   ----------   -8
x1 =	0		x2 =	3

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Ejercicio 18
	-3	x2	+	3	x	+	6	=	0

a = -3
b = 3
c = 6

x =	________________________   - ( 3 ) ± √ ( 3 )2 - 4 ( -3 )( 6 )   -------------------------------------------------------------   2( -3 )
x =	________________________   -3 ± √ 9 + 72   -------------------------------------------------------------   -6
x =	________   -3 ± √ 81     ------------------------------------       -6
x =	-3 ± 9     -----------------------------     -6

x1 =	-3 + 9   --------------     -6		x2 =	-3 - 9   --------------     -6
x1 =	6   ----------   -6		x2 =	-12   ----------   -6
x1 =	-1		x2 =	2

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Ejercicio 19
	-2	x2	-	2	x	+	4	=	0

a = -2
b = -2
c = 4

x =	________________________   - ( -2 ) ± √ ( -2 )2 - 4 ( -2 )( 4 )   -------------------------------------------------------------   2( -2 )
x =	________________________   2 ± √ 4 + 32   -------------------------------------------------------------   -4
x =	________   2 ± √ 36     ------------------------------------       -4
x =	2 ± 6     -----------------------------     -4

x1 =	2 + 6   --------------     -4		x2 =	2 - 6   --------------     -4
x1 =	8   ----------   -4		x2 =	-4   ----------   -4
x1 =	-2		x2 =	1

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Ejercicio 20
	-2	x2	+	6	x	-	4	=	0

a = -2
b = 6
c = -4

x =	________________________   - ( 6 ) ± √ ( 6 )2 - 4 ( -2 )( -4 )   -------------------------------------------------------------   2( -2 )
x =	________________________   -6 ± √ 36 - 32   -------------------------------------------------------------   -4
x =	________   -6 ± √ 4     ------------------------------------       -4
x =	-6 ± 2     -----------------------------     -4

x1 =	-6 + 2   --------------     -4		x2 =	-6 - 2   --------------     -4
x1 =	-4   ----------   -4		x2 =	-8   ----------   -4
x1 =	1		x2 =	2