Solving equations by factoring and solving problems. (pg. 393)
1.
x+3 = 0, so x = -3
3x-4 = 0, so x = 4/3
3.
2x-5 = 0, so x = 5/2
4x+3 = 0, so x = -3/4
5.
(x+8)(x+3) = 0
x+8 = 0, so x = -8
x+3 = 0, so x = -3
7.
(4x-1)(3x+2) = 0
4x-1 = 0, so x = ¼
3x+2 = 0, so x = -2/3
9.
z^2-10z+9 = 0
(z-9)(z-1) = 0
z-9 = 0, so z = 9
z-1 = 0, so z = 1
11.
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.
x^2 – 6x = 8x + x^2
-6x = 8x
x = 0
29.
2x+7 = 0, so x = -7/2
x-10 = 0, so x = 10
31.
3x = 0, so x = 0
x-5 = 0, so x = 5
33.
(x-5)(x+3) = 0
x-5 = 0, so x = 5
x+3 = 0, so x = -3
35.
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.
w^2-5w-36 = 0
(w-9)(w+4) = 0
w-9 = 0, so w = 9
w+4 = 0, so w = -4
39.
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.
5x+1 = 0
So exclude -1/5
5.
3x = 0
So exclude 0
7.
Nothing makes this undefined
9.
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.
9/18 = ½
x^6/x^2 = x^4
y^3/y^5 = 1/y^2
answer:
x^4
-----
2y^2
17.
Factor the top:
8x(1-2x)/8x
Cancel 8x:
1-2x
19.
Factor the top:
(x-3)(x+3)/(x-3)
Cancel x-3:
x+3
21.
Factor:
9(y-2) / 7(y-2)
Cancel y-2:
9/7
23.
Factor:
6(y-3)/2(y-3)
Cancel:
6/2
= 3
25.
Multiply by -1/-1:
-(x-9)/(x-9)
= -1
27.
Factor:
(x+7)(x-7)/(7-x)
Multiply by -1/-1:
-(x+7)(x-7)/(x-7)
= -(x+7)
Multiply and simplify.
39.
Factor:
2(x-2)*6 / 15*-1(x-2)
Cancel x-2:
12/-15
= -4/5
41.
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.
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.
Flip the second one and factor:
2x*5(x+2)
------
5*6(x+2)
Cancel:
10x
-----
30
Simplify:
x/3
57.
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.
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.
Add or subtract, simplify each answer. (pg. 441)
5.
This can’t be simplified and it’s not an addition/subtraction problem.
7.
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.
(x-5)-(x+5)
------
2x
Simplify:
-10
-----
2x
= -5/x
Add or subtract as indicated, simplify.
27.
Common denominator:
8/6x + 9/6x
= 17 / (6x)
29.
Common denominator:
35/14y^2 – 4y/14y^2
(35-4y)/(14y^2)
31.
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.
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.
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.
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.
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.
Common denom:
2x^2/4x + 8/4x = 3x/4x
Clear 4x:
2x^2 + 8 = 3x
2x^2 – 3x + 8 = 0
NO real solutions
41.
3t/4 – 2t/4 = 1
t/4 = 1
t = 4
43.
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.
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.
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.
Common denominator:
3x/6 – 2x/6 = 12
x/6 = 12
x = 72
3.
Cross multiply:
20x = 5*12
20x = 60
X = 60/20
X = 3
5.
Common denominator:
a/a – 4/a = 5
(a-4)/a = 5
a-4 = 5a
-4 = 4a
a = -1
7.
Common denominator:
7a/a + 6/a = 5
(7a+6)/a = 5
7a+6 = 5a
6 = -2a
a = -3
13.
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.
Cross multiply:
2(x-1) = x+1
2x – 2 = x + 1
x-2 = 1
x = 3
19.
Multiply the left by (x+4)/(x+4):
(x+4)/CD = 3x/CD
Clear CD:
x+4 = 3x
2x = 4
X = 2
21.
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.
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.
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.
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.
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.
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.
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