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Find two consecutive even numbers whose product is 624.

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A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.

The height of the banner is 13 cm and the length of the side is 54 cm.

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Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.

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A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.

The image shows a rectangular pathway with a right inlaid in the lower left corner. The right angle of the triangle overlays the lower left corner of the rectangle. The left leg of the right triangle overlays the left side of the rectangle and the hypotenuse of the right triangle runs from the upper left corner of the rectangle to a point on the bottom of the rectangle.

The lengths of the sides of the mosaic are 2.2 and 4.4 feet.

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A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

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The front walk from the street to Pam’s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.

The width of the front walk is 8.1 feet and its length is 30.8 feet.

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For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth. Round answer to the nearest tenth.

The image shows four rectangular tables placed side by side to create one large table.
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A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula h = −16 t 2 + v 0 t to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.

The ball will reach 384 feet on its way up in 4 seconds and on the way down in 6 seconds.

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A bullet is fired straight up from the ground at a velocity of 320 ft/sec. Use the formula h = −16 t 2 + v 0 t to determine when the bullet will reach 800 feet. Round to the nearest tenth.

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10.5 Graphing Quadratic Equations in Two Variables

In the following exercises, graph by plotting point.

In the following exercises, determine if the following parabolas open up or down.

y = −3 x 2 + 3 x 1

down

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y = −4 x 2 7 x + 1

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In the following exercises, find the axis of symmetry and the vertex.

y = x 2 + 6 x + 8

x = 3 ( 3 , 17 )

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In the following exercises, find the x - and y -intercepts.

y = x 2 4 x + 5

y : ( 0 , 5 ) ; x : ( 5 , 0 ) , ( −1 , 0 )

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y = x 2 4 x + 10

y : ( 0 , 10 ) ; x : none

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y = −5 x 2 30 x 46

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y = 16 x 2 8 x + 1

y : ( 0 , 1 ) ; x : ( 1 4 , 0 )

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In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

y = x 2 + 8 x + 15

y : ( 0 , 15 ) ; x : ( −3 , 0 ) , ( −5 , 0 ) ;
axis: x = −4 ; vertex : ( −4 , −1 )
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 17. The parabola has points plotted at the vertex (-4, -1) and the intercepts (-3, 0), (-5, 0) and (0, 15). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -4.

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y = x 2 + 8 x 16

y : ( 0 , −16 ) ; x : ( 4 , 0 ) ;
axis: x = 4 ; vertex : ( 4 , 0 )
This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -15 to 12. The y-axis of the plane runs from -20 to 2. The parabola has points plotted at the vertex (4, 0) and the intercept (0, -16). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.

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y = x 2 + 6 x + 13

y : ( 0 , 13 ) ; x : none ;
axis: x = −3 ; vertex : ( −3 , 4 )
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 18. The parabola has points plotted at the vertex (-3, 4) and the intercept (0, 13). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.

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y = −2 x 2 8 x 12

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y = −4 x 2 + 16 x 11

y : ( 0 , −11 ) x : ( 3.1 , 0 ) , ( 0.9 , 0 ) ;
axis: x = 2 ; vertex: ( 2 , 5 )
This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (2, 5) and the intercepts (3.1, 0) and (0.9, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.

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In the following exercises, find the minimum or maximum value.

y = 7 x 2 + 14 x + 6

The minimum value is −1 when x = −1 .

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y = −3 x 2 + 12 x 10

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In the following exercises, solve. Rounding answers to the nearest tenth.

A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation h = −16 t 2 + 112 t to find how long it will take the ball to reach maximum height, and then find the maximum height.

In 3.5 seconds the ball is at its maximum height of 196 feet.

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A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation A = −2 x 2 + 180 x gives the area, A , of the yard for the length, x , of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

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Practice test

Use the Square Root Property to solve the quadratic equation: 3 ( w + 5 ) 2 = 27 .

w = −2 , −8

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Use Completing the Square to solve the quadratic equation: a 2 8 a + 7 = 23 .

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Use the Quadratic Formula to solve the quadratic equation: 2 m 2 5 m + 3 = 0 .

m = 1 , 3 2

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Solve the following quadratic equations. Use any method.

3 n 2 + 8 n + 3 = 0

n = −4 ± 7 3

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x ( x + 3 ) + 12 = 0

no real solution

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Use the discriminant to determine the number of solutions of each quadratic equation.

6 p 2 13 p + 7 = 0

2

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Solve by factoring, the Square Root Property, or the Quadratic Formula.

Find two consecutive even numbers whose product is 360.

Two consecutive even number are −20 and −18 and 18 and 20.

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The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)

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For each parabola, find which ways it opens, the axis of symmetry, the vertex, the x - and y -intercepts, and the maximum or minimum value.

y = 3 x 2 + 6 x + 8

up x = −1 ( −1 , 5 ) y : ( 0 , 8 ) ; x : none minimum value of 5 when x = −1

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y = x 2 + 10 x + 24

up x = −5 ( −5 , −1 ) y ; ( 0 , 24 ) ; x : ( −6 , 0 ) , ( −4 , 0 ) minimum value of −5 when x = −1

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y = −3 x 2 + 12 x 8

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y = x 2 8 x + 16

down x = −4
( −4 , 32 ) y ; ( 0 , 16 ) ; x : ( −9.7 , 0 ) , ( 1.7 , 0 )
maximum value of 32 when x = −4

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Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry.

y = 16 x 2 + 24 x + 9

y : ( 0 , 9 ) ; x : ( 3 4 , 0 ) ;
axis: x = 3 4 ; vertex : ( 3 4 , 0 )
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (3 fourths, 0) and the intercept (0, 9). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 fourths.

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Solve.

A water balloon is launched upward at the rate of 86 ft/sec. Using the formula h = −16 t 2 + 86 t , find how long it will take the balloon to reach the maximum height and then find the maximum height. Round to the nearest tenth.

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Questions & Answers

integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
Practice Key Terms 6

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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