JustAnswer



Solving equations by factoring and solving problems. (pg. 393)

1. [pic]

x+3 = 0, so x = -3

3x-4 = 0, so x = 4/3

3. [pic]

2x-5 = 0, so x = 5/2

4x+3 = 0, so x = -3/4

5. [pic]

(x+8)(x+3) = 0

x+8 = 0, so x = -8

x+3 = 0, so x = -3

7. [pic]

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

4x-1 = 0, so x = ¼

3x+2 = 0, so x = -2/3

9. [pic]

z^2-10z+9 = 0

(z-9)(z-1) = 0

z-9 = 0, so z = 9

z-1 = 0, so z = 1

11. [pic]

5x^2 + 2x = 3

5x^2 + 2x – 3 = 0

(5x-3)(x+1) = 0

5x-3 = 0, so x = 3/5

x+1 = 0, so x = -1

13. [pic]

x^2 – 6x = 8x + x^2

-6x = 8x

x = 0

29. [pic]

2x+7 = 0, so x = -7/2

x-10 = 0, so x = 10

31. [pic]

3x = 0, so x = 0

x-5 = 0, so x = 5

33. [pic]

(x-5)(x+3) = 0

x-5 = 0, so x = 5

x+3 = 0, so x = -3

35. [pic]

2(6x^2+x-1) = 0

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

3x-1 = 0, so x = 1/3

2x+1 = 0, so x = -1/2

37. [pic]

w^2-5w-36 = 0

(w-9)(w+4) = 0

w-9 = 0, so w = 9

w+4 = 0, so w = -4

39. [pic]

This is a perfect square:

(5x-4)(5x-4) = 0

5x-4 = 0, so x = 4/5

51. An electrician needs to run a cable from the top of a 60-foot tower to a transmitter box located 45 feet away from the base of the tower. Find how long he should cut the cable.

Pythagorean Theorem:

a^2 + b^2 = c^2

c = sqrt(a^2+b^2)

Sqrt(60^2+45^2)

= 75 feet

53. The shorter leg of a right triangle is two feet less then the other leg. Find the length of the two legs if the hypotenuse is 10 feet.

The legs are x and x-2:

x^2 + (x-2)^2 = 10^2

x^2 + x^2 – 4x + 4 = 100

2x^2 – 4x – 96 = 0

x^2 – 2x – 48 = 0

(x-8)(x+6) = 0

x = 8 or -6

x can’t be negative, so:

x = 8, x-2 = 6

8 feet and 6 feet

61. Marie has a rectangular board 12 inches by 16 inches around which she wants to put a uniform border of shells. If she has enough shells for a border whos area is 128 square inches, determine the width of the border.

(2x+12)(2x+16) = 128 + 12*16

4x^2 + 56x – 128 = 0

x^2 + 14x – 32 = 0

(x+16)(x-2) = 0

x = -16 or 2

It can’t be negative, so the width is 2 inches

Find all numbers for which each rational expression is undefined (pg. 431)

3. [pic]

5x+1 = 0

So exclude -1/5

5. [pic]

3x = 0

So exclude 0

7. [pic]

Nothing makes this undefined

9. [pic]

x^3 + x^2-2x = 0

x(x^2+x-2) = 0

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

exclude 0, -2, 1

Simplify each rational expression.

15. [pic]

9/18 = ½

x^6/x^2 = x^4

y^3/y^5 = 1/y^2

answer:

x^4

-----

2y^2

17. [pic]

Factor the top:

8x(1-2x)/8x

Cancel 8x:

1-2x

19. [pic]

Factor the top:

(x-3)(x+3)/(x-3)

Cancel x-3:

x+3

21. [pic]

Factor:

9(y-2) / 7(y-2)

Cancel y-2:

9/7

23. [pic]

Factor:

6(y-3)/2(y-3)

Cancel:

6/2

= 3

25. [pic]

Multiply by -1/-1:

-(x-9)/(x-9)

= -1

27. [pic]

Factor:

(x+7)(x-7)/(7-x)

Multiply by -1/-1:

-(x+7)(x-7)/(x-7)

= -(x+7)

Multiply and simplify.

39. [pic]

Factor:

2(x-2)*6 / 15*-1(x-2)

Cancel x-2:

12/-15

= -4/5

41. [pic]

Factor:

6a(3-2a)*(2a+1)(2a+3)

-------------------------------

(2a+1)(2a+1)*(2a-3)(2a+3)

Cancel:

6a(3-2a)

---------------

(2a+1)(2a-3)

Cancel with a minus sign:

-6a

-------

2a+1

43. [pic]

Factor:

9(x+1)2(x+2)

--------------------

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

Cancel:

9*2

---------

4*3(x-1)

Simplify:

18

-------

12(x-1)

Reduce:

3

2(x-1)

Divide and simplify.

55. [pic]

Flip the second one and factor:

2x*5(x+2)

--------------

5*6(x+2)

Cancel:

10x

-----

30

Simplify:

x/3

57. [pic]

Flip the second and multiply:

(a+b)(4a^3+b)

---------------------

ab(a^2-b^2)

Factor:

(a+b)(4a^3+b)

--------------------

ab(a+b)(a-b)

Cancel:

(4a^3+b)

--------------------

ab(a-b)

59. [pic]

Flip the second and multiply, and factor:

(x-3)(x-3)*4

------------------

(x-3)(x+2)(x-3)(x+3)

Cancel terms:

4

--------------

(x+2)(x+3)

61. [pic]

Add or subtract, simplify each answer. (pg. 441)

5. [pic]

This can’t be simplified and it’s not an addition/subtraction problem.

7. [pic]

I’m guessing that this is addition, not division, since it’s in the addition section…

Add the numerators:

(2x-6)+(3-3x)

------------------

x^2+x-6

Simplify:

(-x-3)

------------

x^2+x-6

Factor:

-(x+3)

------------

(x+3)(x-2)

Cancel:

-1

-----

x-2

9. [pic]

(x-5)-(x+5)

-------------

2x

Simplify:

-10

-----

2x

= -5/x

Add or subtract as indicated, simplify.

27. [pic]

Common denominator:

8/6x + 9/6x

= 17 / (6x)

29. [pic]

Common denominator:

35/14y^2 – 4y/14y^2

(35-4y)/(14y^2)

31. [pic]

Common denom: x^2-16:

(x-3)(x-4)/CD – (x+2)(x+4)/CD

= (x^2-7x+12)/CD – (x^2+6x+8)/CD

= (x^2-7x+12-x^2-6x-8)/CD

= (-13x+4)/(x^2-16)

33. [pic]

Common denom = x^2-x-20

(x+4)/CD + (2x-19)/CD

= (3x-15)/CD

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

= 3/(x+4)

43. [pic]

Common denom: (y+4)(y-4)(y-2)

(y+1)(y+4)/CD – 3(y-2)/CD

= (y^2+5y+4)/CD – (3y-6)/CD

= (y^2+5x+4-3y+6)/CD

= (y^2+2y+10)/CD

= (y^2+2y+10)/((y+4)(y-4)(y-2))

45. [pic]

I’m assuming that the first one is x^2, not x^3.

Common denom = (x-2)(x+1)(x+3)

7(x+3)/CD + x(x-2)/CD

(7x + 21 + x^2 – 2x)/CD

(x^2+5x+21)/CD

(x^2+5x+21)/((x-2)(x+1)(x+3))

47. [pic]

Factor the bottoms: (3x+2)(x+3) and (x+3)(2x-5)

Common denom = (3x+2)(x+3)(2x-5)

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

(2x^2+3x-20)/CD + (3x^2+2x)/CD

(5x^2+5x-20)/CD

5(x^2+x-4)/CD

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

Solve each equation and Check. (pg.88)

39. [pic]

Common denom:

2x^2/4x + 8/4x = 3x/4x

Clear 4x:

2x^2 + 8 = 3x

2x^2 – 3x + 8 = 0

NO real solutions

41. [pic]

3t/4 – 2t/4 = 1

t/4 = 1

t = 4

43. [pic]

Common denom of 28:

7(n-3)/28 + 4(n+5)/28 = 10/28

Cancel 28:

7n – 21 + 4n + 20 = 10

11n – 1 = 10

11n = 11

n = 1

check: (1-3)/4 + (1+5)/7 = -2/4 + 6/7 = 5/14, yes!

47. [pic]

Split:

3x/9 – 1/9 + x = 3x/3 + 1/3 + 4

4/3 x – 1/9 = x + 13/3

1/3 x – 1/9 = 13/3

1/3 x = 40/9

x = 40/3

check: (3*40/3-1)/9 + 40/3 = 53/3

(3*40/3+1)/3 + 4 = 53/3

69. [pic]

Multiply by 15:

5(m-4)-3(3m-1) = 15

5m-20-9m+3 = 15

-4m-17 = 15

-4m = 32

m = -8

check: (-8-4)/3-(3*-8-1)/5 = 1, yes!

Solve Each Equation. (pg. 457)

1. [pic]

Common denominator:

3x/6 – 2x/6 = 12

x/6 = 12

x = 72

3. [pic]

Cross multiply:

20x = 5*12

20x = 60

X = 60/20

X = 3

5. [pic]

Common denominator:

a/a – 4/a = 5

(a-4)/a = 5

a-4 = 5a

-4 = 4a

a = -1

7. [pic]

Common denominator:

7a/a + 6/a = 5

(7a+6)/a = 5

7a+6 = 5a

6 = -2a

a = -3

13. [pic]

Get the CD (x^2+2x-8) on the left terms, and cancel the CD from all the terms:

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

5x+20-2x+4 = -4

3x+24 =-4

3x = -28

x = -28/3

15. [pic]

Cross multiply:

2(x-1) = x+1

2x – 2 = x + 1

x-2 = 1

x = 3

19. [pic]

Multiply the left by (x+4)/(x+4):

(x+4)/CD = 3x/CD

Clear CD:

x+4 = 3x

2x = 4

X = 2

21. [pic]

Common denom on the left is x^2-2x:

x/(x^2-2x) – 2/(x^2-2x) = 1

Multiply by x^2-2x:

x-2 = x^2-2x

x^2-3x+2 = 0

x = 1 or 2

2 makes the original equation undefined, so:

X = 1

23. [pic]

Convert to common denom: 6x^2+6x

(3x+3)/CD – 6x/CD = 2/CD

Cancel CD:

3x+3-6x=2

-3x+3=2

-3x=-1

X = 1/3

29. [pic]

Multiply both sides by x+2:

x+3 = 1

Subtract 3:

X = -2

However, that’s a false solution since it makes x+2 = 0, and the fraction undefined.

NO solution

31. [pic]

Common denom: a^2-9

(a+3)/CD + 2(a-3)/CD = 1/CD

Clear CD:

a+3+2a-6 = 1

3a-3 = 1

3a = 4

a= 4/3

33. [pic]

Multiply by x^2-9:

64 + x^2-9 = 2x(x-3)

64 + x^2 – 9 = 2x^2 – 6x

x^2 – 6x - 55 = 0

(x-11)(x+5) = 0

X = 11 or -5

35. [pic]

Multiply by 4y+1:

-15 + 4(4y+1) = y(4y+1)

-15 + 16y + 4 = 4y^2 + y

4y^2 – 15y + 11 = 0

(x-1)(4x-11) = 0

x = 1 or 11/4

39. [pic]

Common denominator

3(x+2)/CD = (x+5)/CD – (3x+6)/CD

Cancel CD:

3(x+2) = (x+5) – (3x+6)

3x + 6 = x + 5 – 3x – 6

3x + 6 = -2x – 1

5x + 6 = -1

5x = -7

x = -7/5

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