Factoring
Factor: x2 + 4x +4
2 * 2 = 4
2 + 2 = 4
Answer: (x + 2) (x + 2) or (x + 2)2
Solve for x:
y = 5x + 2
y = 3x + 8
y = y, so 5x + 2 = 3x -8
-3x -2 -3x -2
2x = -6
2x / 2 = -6 /2
Answer: x = -3
Rewrite the equation so its in the y= form:
5x - 2y = 8
5x - 2y = 8
-5x -5x
-2y = -5x + 8
/-2 /-2
Answer: y = 2.5x - 4
Your family plans to buy a new air conditioner. They can buy the Super Cool X1400 for $800, or they can buy the Efficient Energy X2000 for $1200. Both models will cool your home equally well, but the Efficient Energy model is less expensive to operate. The Super Cool X1400 will cost $60 per month to operate, while the Efficient Energy X2000 costs only $40 per month to operate.
Write an equation to represent the cost of buying and operating the Super Cool X1400 and an equation to represent the cost of buying and operating the Efficient Energy X2000 where C= cost and m= months.
C = 60m + 800
C = 40m + 1200
Find the inverse equation:
f(x) = (x - 4)2
x = (y - 4)2
Sqrt(x) = sqrt((y - 4)2)
Sqrt(x) = y - 4
+4 +4
Answer: f-1(x) = sqrt(x) + 4
Factor: x2 + 5x - 14
7 * -2 = -14
7 + -2 = 5
Answer: (x + 7) (x - 2)
Solve for y:
2x - 5y = 36
x - 3y = 20
-2 (x - 3y = 20) = -2x + 6y = -40
2x - 5y = 36
-2x + 6y = -40
The xs cancel out and you add the common factors
Answer: y = -4
Rewrite the equation so its in the y= form:
2x = 2y - 6
2x = 2y - 6
+6 +6
2x +6 = 2y
/2 /2
Answer: y = x + 3
Leadfoot Lilly was driving 80 miles per hour when she passed a parked highway patrol car. By the time she was half a mile past the spot where the patrol car was parked, the officer was driving after her at 100 miles per hour. If these rates remain constant, how long will it take the officer to catch up to Lilly? Write and solve an equation to represent this situation.
L = 80m
O = 100m + 1/2
80m = 100m + 1/2
20m = 1/2
/20 /20
m = 0.025
It will take the officer 2.5 seconds to catch up with lily.
Find the inverse equation:
f(x) = 3x - 5
x = 3y - 5
+5 +5
x + 5 = 3y
/3 /3
Answer: f-1(x) = (x + 5) / 3
Factor: 2x2 - 6x - 20
DAILY DOUBLE!!!
(2x + 4) (x - 5)
Solve for x:
y = 5x - 4
3x - 2y = 15
3x - 2(5x - 4) = 15
3x - 10x + 8 = 15
-9 -9
-7x = 7
/-7 /-7
Answer: x = -1
Rewrite the equation so its in the y= form:
-x = -y - 3
-x = -y - 3
+3 +3
-x + 3 = -y
/-1 /-1
Answer: y = x - 3
A cell phone company charges a monthly rate of $12.95 and $0.25 a minute per call. The bill for m minutes is $21.20. How many minutes were charged on this bill?
F(m)= .25m + 12.95
21.20 = .25m + 12.95
-12.95 -12.95
8.25 = .25m
/.25 /.25
m = 33
* 33 minutes were charged on the bill
Find the inverse equation:
f(x) = 2(x3 - 5)
x = 2(y3 - 5)
/2 /2
cbrt(x/2) = cbrt(y3 - 5)
cbrt(x/2) = y - 5
+ 5 + 5
Answer: f-1(x) = cbrt(x/2) + 5
Factor: x2 + 2.25
Since there is no middle number, the two numbers must be the same.
sqrt(2.25) = 1.5
Answer: (x + 1.5) (x - 1.5)
Solve for x and y:
5x - 2y = 15
y = 2x - 5
5x - 2(2x - 5) = 15
5x - 4x + 10 = 15
- 10 -10
x = 5
y = 2x - 5
y = 2(5) - 5
y = 10 - 5 Answer: (5, 5)
Rewrite the equation so its in the y= form:
xy + 3x = 2
xy + 3x = 2
/ x /x
y + 3 = 2/x
-3 -3
Answer: y = 2/x - 3
Your math class wants to collect money for a field trip, so it decides to sell two kinds of candy bags. The Chocolate Lovers Bag costs $4.25 for five chocolate truffles and two caramel turtle candies. The Combusting Caramel Bag costs $3.50 for eight caramel turtle candies and two chocolate truffles. How much does each chocolate truffle and caramel turtle candy cost?
-4(4.25 = 5c + 2t)
3.50 = 8t + 2c
-17 = -20c - 8t
-17 = -20c
/-20 /-20
c = .85
4.25 = 5(.85) + 2t
4.25 = 4.25 + 2t
-4.25 -4.25
2t = 0
/2 /2
The chocolate truffles are 85 cents each and the Caramel turtles are free (somehow)
Find the inverse function:
f(x) = 10(x - 4) / 3
x = 10(y - 4) / 3
*3 *3
3x = 10(y - 4)
/10 /10
3x/10 = y - 4
+ 4 + 4
Answer: f-1(x) = 3x/10 + 4
Factor: x2 + 29x + 180
(Use the Quadratic Formula)
-b +/- sqrt(b2 - 4ac) / 2a
-29 +/- sqrt( 292 - 4 (1)(180)) / 2(1)
-29 +/- sqrt(121) / 2
-29 +/- 11 /2
-29 + 11 / 2 = -9
-29 - 11 / 2 = -20
Answer: (x - 9) (x - 20)
Solve for x=
y2 = -4x
y = x + 2
-4x = x2 + 4
x2 + 4x + 4 = 0
(x + 2) (x + 2) = 0
Answer: x = -2
Rewrite the equation so its in the y= form:
xy = 2x
xy = 2x
/x /x
Answer: y = 2
* Not all answers have to have both variables in them.
Jenna and her friend, Khalil, are having a contest to see who can save the most money. Jenna has already saved $110 and every week she saves an additional $20. Khalil has already saved $80 and every week he saves an additional $25. Let x represent the number of weeks and y represent the total amount of money saved. Determine in how many weeks Jenna and Khalil will have the same amount of money.
J: y = 20x + 110
K: y = 25x + 80
20x + 110 = 25x + 80
-25x -110 -25x -110
-5x = -30
/-5 /-5
x = 6
* They will have the same amount in 6 weeks.
Prove these equations are inversed.
f(x) = sqrt(x - 2) + 3
f(g) = (x - 3)2 + 2
sqrt((x - 3)2 +2) -2) +3
(x - 3) + 2) - 2) + 3)
x
Answer: yes, they are inverses!