Please Show Your Work
#1:
Expand the following expression, and reduce (not factor) to its simplest form:
(x + 2) (x – 2) (x + 12)
= (x2 - 4)(x + 12)
= x3 +12x2 - 4x - 48
#2:Expand the following expression, and reduce (not factor) to its simplest form:
(x^2 + 2y) (–2y + x^2)
= (x2 + 2y)(x2 - 2y)
= x4 - 4y2
#3:
Factor the following expression:
x^2 + 10x + 21.
= (x + 3)(x + 7)
#4:
Factor the following expression:
2x^2 – x – 3.
= (2x - 3)(x + 1)
#5:
Factor and simplify the following expression:
x^2 + 16yx + 64y^2
= (x + 8y)(x + 8y)
#6:
Solve for x:
15x – 12 = 7x + 46
15x - 7x - 12 = 7x - 7x + 46
8x - 12 + 12 = 46 + 12
8x = 58
x = 58/8 = 29/4
#7:
Solve for x:
4x^2 – 2x + 5 = 4x^2 + x + 1
-2x + 5 = x + 1
-2x + 2x + 5 = x + 2x + 1
5 = 3x + 1
5 - 1 = 3x + 1 - 1
3x = 4
x = 4/3
#8:
Solve for x:
(x+3)^2 = 225
(x + 3) = ±√225
x + 3 = ±15
x = ± 15 - 3
x = 12, -18
#9:
Solve for x:
x^2 + 6x + 6 = 118
x 2 + 6x + 6 "118 = 0
x 2 + 6x "112 = 0
( x +14 )( x " 8) = 0
x = 8, "14
#10:
!
Solve for x:
x^2 + 8x + 15 = 0
( x + 5)( x + 3) = 0
x = "3, " 5
#11:
!
Solve for x:
3x^2 + 10x + 3 = 0
(3x +1)( x + 3) = 0
x = "1/3, " 3
!
#12:
Add and express the result in its simplest form:
1
2
+
x+4 x
=
x
2
+
x ( x + 4) x
=
x
2( x + 4 )
+
x ( x + 4) x ( x + 4 )
=
x + 2x + 8
x ( x + 4)
=
3x + 8
x ( x + 4)
#13:
!
Add and express the result in its simplest form:
2
1
"
x+4 x+3
!
=
2( x + 3)
1
"
( x + 4 )( x + 3) x + 3
=
2( x + 3)
x+4
"
( x + 4 )( x + 3) ( x + 4)( x + 3)
=
2x + 6 " x " 4
( x + 4 )( x + 3)
=
x +2
( x + 4 )( x + 3)
#14:
Add and express the result in its simplest form:
2
1
3
"
+
x+4 x+3 x
=
2x ( x + 3)
1
3
"
+
x ( x + 4 )( x + 3) x + 3 x
=
2x ( x + 3)
x ( x + 4)
3
"
+
x ( x + 4 )( x + 3) x ( x + 4 )( x + 3) x
=
2x ( x + 3)
x ( x + 4)
3( x + 4 )( x + 3)
"
+
x ( x + 4 )( x + 3) x ( x + 4 )( x + 3) x ( x + 4 )( x + 3)
=
2x ( x + 3) " x ( x + 4 ) + 3( x + 4 )( x + 3)
x ( x + 4 )( x + 3)
=
2x 2 + 6x " x 2 " 4 x + 3x 2 + 21x + 36
x ( x + 4 )( x + 3)
4 x 2 + 23x + 36
=
x ( x + 4 )( x + 3)
#15:
!
Multiply and express the result in its simplest form:
2x
1
# 3
x "3 x
!
=
2x
x ( x " 3)
=
2
x ( x " 3)
3
2
#16:
Multiply and express the result in its simplest form:
x + 3 2( x " 3)
#
x2 " 9
x
!
=
2( x " 3)( x + 3)
x ( x " 3)( x + 3)
=
2
x
#17:
Divide and express the result in its simplest form (the symbol between the two rational
expressions is dividing):
2x
x +5
÷
x + 5 x +1
=
2x
x +1
"
x +5 x +5
=
2x ( x +1)
( x + 5) 2
#18:
!
Divide and express the result in its simplest form (the symbol between the two rational
expressions is dividing)
3x 2 x " 2
÷
x + 2 x +1
!
=
3x 2 x +1
#
x +2 x "2
=
3x 2 ( x +1)
x2 " 4
#19:
Solve for x:
3
x
"
=0
"5 " x x " 3
3
x
=
"5 " x x " 3
3( x " 3) = x ( "5 " x )
3x " 9 = "5x " x 2
x 2 + 3x + 5x " 9 = 0
x 2 + 8x " 9 = 0
( x + 9)( x "1) = 0
x = 1, " 9
!
#20:
Solve for x:
2x
1
"
=0
x+4 x "3
2x
1
=
x+4 x "3
2x ( x " 3) = x + 4
2x 2 " 6x = x + 4
2x 2 " 7x " 4 = 0
(2x +1)( x " 4 ) = 0
x = 4, "1/2
!
#21:
Jack bought a new digital camera as a birthday present for his wife, but by the time he
needed to record the purchase in his checkbook register, he forgot how much he had paid
for it (let’s just call this amount "x"). He only remembered one small detail: six (6) less
than the square of this amount is equal to the amount itself. How much did he pay for the
camera?
x2 " 6 = x
x2 " x " 6 = 0
( x " 3)( x + 2) = 0
x = 3 (the negative solution is discarded)
!
#22:
Given: x>0
Find x, such that:
x^2 – 25 > x + 5
x 2 " x " 25 " 5 > 0
x 2 " x " 30 > 0
( x " 6)( x + 5) > 0
With both terms positive :
x " 6 > 0, and x + 5 > 0
x > 6, and x > "5
x >6
There is also a solution for both terms being negative, but the
result is x < -5 and the problem statement specified x > 0
!