Question

Algebra de Mancil tomo 2 pag 122 ejercicios 148.
Resolver las ecuaciones siguientes por descomposición en factores

Answer 1

Hola! 

1) (x-3)(x+7) = 0    2) (x-6)(x+5) = 0   3) (y-2)(y+4) = 0  4) (y+5)(y+12) = 0

5) (x-1)(x+8) = 0    6) (z-10)(z-3) = 0     7) 2*(x - 3/2)(x - 1) = 0 

8) 2*(x-4)(x + 3/2) = 0   9) 6*(y- 3/2)(y + 2/3) = 0   10) 4*(t - 3/4)(t + 3) = 0

11) 4*(x - 3/2)(x - 3/2) = 0   12) 6*(x - 5/2)(x + 4/3) = 0 

13) 8*(x - 1/2)(x + 3/4) = 0   14) 3*(y - 7/3)(y+2) = 0

15) 30*(x - 1/5)(x + 5/6) = 0    16) 14*(x - 3/2)(x + 4/7) = 0

Espero haberte ayudado.

Answer 2

Al resolver las ecuaciones por descomposición en factores se obtiene:

1. (x-3)(x+7)

2. (x-6)(x+5)

3. (y+2)(y+4)

4. (y+5)(y+12)

5. (x-1)(x+8)  

6. (z-10)(z+3)

7. (x-3/2)(x+1)

8. (x-4)(x+3/2)

9. (y-3/2)(y+2/3)

10. (t-3/4)(t+3)

11. (z-3/2)(z-3/2)

12. (x-5/2)(x+4/3)

13. (x-1/2)(x+3/4)

14. (y-7/3)(y+2)

15.(x-1/5)(x+5/6)

16. (z-3/2)(z+4/7)

Explicación:

Los polinomios de grado 2, se calculan sus raíces con la formula resolvente;

$x_{1} = \frac{-b+\sqrt{b^{2}-4ac } }{2a}$

$x_{2} = \frac{-b-\sqrt{b^{2}-4ac } }{2a}$

1. x² + 4x - 21 = 0

Siendo;

a = 1; b = 4; c = -21

Sustituir;

$x_{1} = \frac{-4+\sqrt{4^{2}-4(-21) } }{2}$

$x_{1} = \frac{-4+\sqrt{100 } }{2}$

$x_{1} = \frac{-4+10 }{2}$

x₁ = 3

$x_{2} = \frac{-4-10 }{2}$

x₂ = -7

=(x-3)(x+7)

2. x² - x - 30 = 0

Siendo;

a = 1; b = -1; c = -30

Sustituir;

$x_{1} = \frac{1+\sqrt{-1^{2}-4(-30) } }{2}$

$x_{1} = \frac{1+\sqrt{121 } }{2}$

$x_{1} = \frac{1+11 }{2}$

x₁ = 6

$x_{2} = \frac{1-11 }{2}$

x₂ = -5

=(x-6)(x+5)

3. y² +2y  - 8 = 0

Siendo;

a = 1; b = 2; c = -8

Sustituir;

$y_{1} = \frac{-2+\sqrt{2^{2}-4(-8) } }{2}$

$y_{1} = \frac{-2+\sqrt{36} }{2}$

$y_{1} = \frac{-2+6 }{2}$

x₁ = -2

$y_{2} = \frac{-2-6 }{2}$

y₂ = -4

=(y+2)(y+4)

4. y² + 17y + 60 = 0

Siendo;

a = 1; b = 17; c = 60

Sustituir;

$y_{1} = \frac{-17+\sqrt{17^{2}-4(60) } }{2}$

$y_{1} = \frac{-17+\sqrt{49} }{2}$

$y_{1} = \frac{-17+7 }{2}$

x₁ = -5

$y_{2} = \frac{-17-7 }{2}$

y₂ = -12

=(y+5)(y+12)

5. x² + 7x - 8 = 0

Siendo;

a = 1; b = 7; c = -8

Sustituir;

$x_{1} = \frac{-7+\sqrt{7^{2}-4(-8) } }{2}$

$x_{1} = \frac{-7+\sqrt{81} }{2}$

$x_{1} = \frac{-7+9 }{2}$

x₁ = 1

$x_{2} = \frac{-7-9 }{2}$

x₂ = -8

=(x-1)(x+8)

6. z² - 7z - 30 = 0

Siendo;

a = 1; b = -7; c = -30

Sustituir;

$z_{1} = \frac{7+\sqrt{-7^{2}-4(-30) } }{2}$

$z_{1} = \frac{7+\sqrt{169} }{2}$

$z_{1} = \frac{7+13 }{2}$

z₁ = 10

$z_{2} = \frac{7-13 }{2}$

z₂ = -3

=(z-10)(z+3)

7. 2x² - x - 3 = 0

Siendo;

a = 2; b = -1; c = -3

Sustituir;

$x_{1} = \frac{1+\sqrt{-1^{2}-4(2)(-3) } }{4}$

$x_{1} = \frac{1+\sqrt{25} }{4}$

$x_{1} = \frac{1+5 }{4}$

x₁ = 3/2

$x_{2} = \frac{1-5 }{4}$

x₂ = -1

=(x-3/2)(x+1)

8. 2x² - 5x - 12 = 0

Siendo;

a = 2; b = -5; c = -12

Sustituir;

$x_{1} = \frac{5+\sqrt{-5^{2}-4(2)(-12) } }{4}$

$x_{1} = \frac{5+\sqrt{121} }{4}$

$x_{1} = \frac{5+11 }{4}$

x₁ = 4

$x_{2} = \frac{5-11 }{4}$

x₂ = -3/2

=(x-4)(x+3/2)

9. 6y² - 5y - 6 = 0

Siendo;

a = 6; b = -5; c = -6

Sustituir;

$y_{1} = \frac{5+\sqrt{-5^{2}-4(6)(-6) } }{12}$

$y_{1} = \frac{5+\sqrt{169} }{12}$

$y_{1} = \frac{5+13 }{12}$

x₁ = 3/2

$y_{2} = \frac{5-13 }{12}$

y₂ = -2/3

=(y-3/2)(y+2/3)

10. 4t² + 9t - 9 = 0

Siendo;

a = 4; b = 9; c = -9

Sustituir;

$t_{1} = \frac{-9+\sqrt{9^{2}-4(6)(-6) } }{8}$

$t_{1} = \frac{-9+\sqrt{225} }{8}$

$t_{1} = \frac{-9+15 }{8}$

t₁ = 3/4

$t_{2} = \frac{-9-15 }{8}$

t₂ = -3

=(t-3/4)(t+3)

11. 4z² - 12z + 9 = 0

Siendo;

a = 4; b = -12; c = 9

Sustituir;

$z_{1} = \frac{12+\sqrt{-12^{2}-4(-30) } }{8}$

$z_{1} = \frac{12+0 }{8}$

z₁ = 3/2

$z_{2} = \frac{12}{8}$

z₂ = 3/2

=(z-3/2)(z-3/2)

12. 6x² - 7x - 20 = 0

Siendo;

a = 6; b = -7; c = -20

Sustituir;

$x_{1} = \frac{7+\sqrt{-7^{2}-4(2)(-12) } }{12}$

$x_{1} = \frac{7+\sqrt{529 }{12}$

$x_{1} = \frac{7+23}{12}$

x₁ = 5/2

$x_{2} = \frac{7-23 }{12}$

x₂ = -4/3

=(x-5/2)(x+4/3)

13. 8x² + 2x - 3 = 0

Siendo;

a = 8; b = 2; c = -3

Sustituir;

$x_{1} = \frac{-2+\sqrt{2^{2}-4(2)(-12) } }{16}$

$x_{1} = \frac{-2+\sqrt{100 }{16}$

$x_{1} = \frac{-2+10}{16}$

x₁ = 1/2

$x_{2} = \frac{-2-10 }{16}$

x₂ = -3/4

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

14. 3y² - y - 14 = 0

Siendo;

a = 3; b = -1; c = -14

Sustituir;

$y_{1} = \frac{1+\sqrt{-1^{2}-4(3)(-14) } }{6}$

$y_{1} = \frac{1+\sqrt{169} }{6}$

$y_{1} = \frac{1+13 }{6}$

x₁ = 7/3

$y_{2} = \frac{1-13 }{6}$

y₂ = -2

=(y-7/3)(y+2)

15. 30x² + 19x - 5 = 0

Siendo;

a = 30; b = 19; c = -5

Sustituir;

$x_{1} = \frac{-19+\sqrt{19^{2}-4(30)(-5) } }{60}$

$x_{1} = \frac{-19+\sqrt{961 }{60}$

$x_{1} = \frac{-19+31}{60}$

x₁ = 1/5

$x_{2} = \frac{-19-31 }{60}$

x₂ = -5/6

=(x-1/5)(x+5/6)

11. 14z² - 13z - 12 = 0

Siendo;

a = 14; b = -13; c = -12

Sustituir;

$z_{1} = \frac{13+\sqrt{-13^{2}-4(-30) } }{28}$

$z_{1} = \frac{13+sqrt{841} }{28}$

$z_{1} = \frac{13+29 }{28}$

z₁ = 3/2

$z_{2} = \frac{12}{28}$

z₂ = -4/7

=(z-3/2)(z+4/7)

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