Ecuaciones de Primer grado con una variable

Ecuaciones de Primer grado con una variable

1. Resolver las siguientes ecuaciones:

1. 5x - 15 = x + 25

5x - 15 = x+25

5x – x = 25 + 15

4x = 25 + 15

4x = 40

x = 10

2. 3(x - 5) + 6 = 4(x + 7) – 39

3x – 15 + 6 = 4 (x+7) - 39

3x – 9 = 4 (x+7) – 39

3x – 9 = 4x + 28 -39

3x – 9 = 4x – 11

3x – 4x = -11 + 9

-x = -11+ 9

-x = -2

x= 2

3. (x + 6)(x - 5) = x2 + 3x – 53

(x+6) (x-5) - x2 – 3x = -53

x2 – 5x + 6x – 30 - x2 – 3x = -53

x2 + x -30 - x2 – 3x = -53

x-30-3x = -53

-2x – 30 = -53

-2x = -53 + 30

-2x = -23

x=

x=

x= 11,5

4. x2 + 6x + 10 = (x + 8)2

x2 + 6x - (x + 8)2 = -10

x2 + 6x – (x2 + 16x + 82) = -10

x2 + 6x – (x2+ 16x + 64) = -10

x2 + 6x – x2 -16x – 64 = -10

6x – 16x -64 = -10

-10x -64 = -10

-10x = -10 + 64

-10x = 54

x= -

x=

x= -5,4

5. 3(x + 2) + 4(2x - 1)2 = 16x2 + 3x – 22

3x + 6 + 4 (2x - 1)2 = 16x2 + 3x – 22

3x + 4 (2x - 1)2 - 16x2 – 3x = -22-6

4 (2x - 1)2 - 16x2 = -22-6

4 (2x - 1)2 - 16x2 = -28

4 (4x2 – 4x + 1) - 16x2 = -28

16x2 - 16x + 4 -16x2 = -28

- 16x + 4 = -28

-16x = -28 – 4

-16x = -32

x=

x= 2

Ecuaciones de Segundo grado

Resolver las siguientes ecuaciones:

01) x2 = 81

x =

x =

x = 9; -9

02) 14x2 - 28 = 0

14x2 = 28

x2 =

x2 = 2

x =

x = ; -

03) (x + 6)(x - 6) = 13

x2 - 62 = 13

x2 – 36 = 13

x2 = 13 + 36

x2 = 49

x =

x =

x = 7; -7

04) (2x - 5)(2x + 5) - 119 = 0

(2x) 2 - 52 - 119 = 0

4x2- 52 - 119 = 0

4x2- 25 -119 = 0

4x2 -144 = 0

4x2 = 144

x2 =

x2 = 36

x =

x =

x = 6; -6

05) (x + 11)(x - 11) = 23

x2 -112 = 23

x2 = 23 + 112

x2 = 23 + 121

x2 = 144

x =

x =

x = 12; -12

06) x2 = 7x

x2 -7x = 0

xx – 7x = 0

xx+ x.-7 = 0

x (x-7) = 0

x= 0

x – 7 = 0

x= 7

x = 0; 7

07) 21x2 + 100 = - 5

21x2 = -5 – 100

21x2 = - 105

x2 =

x2 = -5

x =

x = ;

08) 2x2 - 6x = 6x2 - 8x

2x2 - 6x - 6 x2 + 8x = 0

-4x2 - 6x + 8x = 0

-4x2 + 2x = 0

x (-4x + 2) = 0

x = 0;

09) (x - 3)2 - (2x + 5)2 = - 16

(x – 3 – (2x + 5)) (x-3 + 2x + 5) = -16

(x – 3 – (2x + 5)) ( 3x – 3 + 5) = -16

(x – 3 – (2x + 5)) ( 3x + 2) = -16

(x – 3 – 2x -5 ) ( 3x + 2) = -16

(-x -3 -5) ( 3x + 2) = -16

(-x -8 ) (3x + 2) = -16

-3x2 - 2x – 24x -16 = -16

-3x2 - 26x -16 = -16

-3x2 - 26x = -16 + 16

-3x2 - 26x = 0

x (-3x – 26 ) = 0

x = 0;

10) (4x - 1)(2x + 3) = (x + 3)(x - 1)

(4x - 1)(2x + 3) - (x + 3)(x - 1) = 0

8x2 + 12x – 2x -3 - (x + 3)(x - 1) = 0

8x2 + 10x -3 - (x + 3)(x - 1) = 0

8x2 + 10x - (x + 3)(x - 1) = 3

8x2 + 10x – (x2 –x + 3x -3) = 3

8x2 + 10x – (x2 + 2x -3) = 3

8x2 + 10x – x2 - 2x -3 = 3

7x2 + 10x – 2x + 3 = 3

7x2 + 8x + 3 = 3

7x2 + 8x = 3 – 3

7x2 + 8x = 0

x (7x + 8) = 0

x= 0;

11) x2 + 12x + 35 = 0

En este ejercicio preferí factorizar a fin de hallar un par de números que me ayudaran a resolver esta ecuación, esto lo logro buscando dos números que sumados me den 12 y multiplicándolos me den 35, estos son:

5 y 7

5+7=12

5.7= 35

Entonces la ecuación queda:

(x+5) (x+7) = 0

x + 5 = 0

x + 7 = 0

x = -5

x = -7

x = -5; -7

12) x2 - 3x + 2 = 0

Al igual que en el ejercicio anterior, factoricé a fin de resolver la ecuación. Los dos números encontrados son:

-1 y -2

-1-2 = -3

(-1) (-2) = 2

(x-1) (x-2) = 0

x-1 = 0

x-2= 0

x = 1

x = 2

x = 1; 2

13) x2 + 4x =285

Aquí decidí simplificar y luego factorizar

x2 + 4x - 285 =285 -285

x2 + 4x - 285 = 0

Al factorizar encontré los números:

19 y -15

19 -15 = 4

(19) (-15) = -285

(x + 19) (x -15) = 0

x + 19 = 0

x – 15 = 0

x = -19

x= 15

x= -19; 15

14) 5x(x - 1) - 2(2x2 - 7x) = - 8

5x2 – 5x - 4x2 + 14x

5x2 - 4x2 + 14x – 5x

5x2 - 4x2 + 9x

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