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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

how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
find the solution to the following functions, check your solutions by substitution. f(x)=x^2-17x+72
Carlos Reply
Aziza is solving this equation-2(1+x)=4x+10
Sechabe Reply
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
KAJAL Reply
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
Adedamola Reply
please the answer to the example exercise
Patricia Reply
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
.473
Angelita
?
Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
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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