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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

What is the lcm of 340
Kendra Reply
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
Quadratic equations involving factorization
Winner Reply
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x
Sandra Reply
What is c+4=8
Penny Reply
2
Letha
4
Lolita
4
Rich
4
thinking
C+4=8 -4 -4 C =4
thinking
I need to study
Letha
4+4=8
William
During two years in college, a student earned $9,500. The second year, she earned $500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp
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.
Josh Reply
Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
Ida Reply
hey
Juan
Sup
patrick
The sum of two numbers is 155. The difference is 23. Find the numbers
Michelle Reply
The sum of two numbers is 155. Their difference is 23. Find the numbers
Michelle
The difference between 89 and 66 is 23
Ciid
Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
Amber Reply
hello
bismark
I need a math tutor BAD
Stacie
Me too
Letha
me too
Xavier
ok
Bishal
teet
Bishal
my answer is y10 that right
misty
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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