# 10.5 Graphing quadratic equations  (Page 9/15)

 Page 9 / 15

Find two consecutive even numbers whose product is 624.

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.

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.

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

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.

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.

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.

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.

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.

## 10.5 Graphing Quadratic Equations in Two Variables

In the following exercises, graph by plotting point.

Graph $y={x}^{2}-2$

Graph $y=\text{−}{x}^{2}+3$

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

$y=-3{x}^{2}+3x-1$

down

$y=5{x}^{2}+6x+3$

$y={x}^{2}+8x-1$

up

$y=-4{x}^{2}-7x+1$

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

$y=\text{−}{x}^{2}+6x+8$

$x=3$ $\left(3,17\right)$

$y=2{x}^{2}-8x+1$

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

$y={x}^{2}-4x+5$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,5\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(5,0\right),\left(-1,0\right)$

$y={x}^{2}-8x+15$

$y={x}^{2}-4x+10$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,10\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none}$

$y=-5{x}^{2}-30x-46$

$y=16{x}^{2}-8x+1$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,1\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(\frac{1}{4},0\right)$

$y={x}^{2}+16x+64$

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

$y={x}^{2}+8x+15$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,15\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-3,0\right),\left(-5,0\right);$
axis: $x=-4;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-4,-1\right)$

$y={x}^{2}-2x-3$

$y=\text{−}{x}^{2}+8x-16$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-16\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(4,0\right);$
axis: $x=4;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(4,0\right)$

$y=4{x}^{2}-4x+1$

$y={x}^{2}+6x+13$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,13\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none};$
axis: $x=-3;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-3,4\right)$

$y=-2{x}^{2}-8x-12$

$y=-4{x}^{2}+16x-11$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-11\right)\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(3.1,0\right),\left(0.9,0\right);$
axis: $x=2;\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(2,5\right)$

$y={x}^{2}+8x+10$

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

$y=7{x}^{2}+14x+6$

The minimum value is $-1$ when $x=-1$ .

$y=-3{x}^{2}+12x-10$

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 $h=-16{t}^{2}+112t$ 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.

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 $A=-2{x}^{2}+180x$ 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.

## Practice test

Use the Square Root Property to solve the quadratic equation: $3{\left(w+5\right)}^{2}=27$ .

$w=-2,-8$

Use Completing the Square to solve the quadratic equation: ${a}^{2}-8a+7=23$ .

Use the Quadratic Formula to solve the quadratic equation: $2{m}^{2}-5m+3=0$ .

$m=1,\frac{3}{2}$

Solve the following quadratic equations. Use any method.

$8{v}^{2}+3=35$

$3{n}^{2}+8n+3=0$

$n=\frac{-4±\sqrt{7}}{3}$

$2{b}^{2}+6b-8=0$

$x\left(x+3\right)+12=0$

no real solution

$\frac{4}{3}{y}^{2}-4y+3=0$

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

$6{p}^{2}-13p+7=0$

2

$3{q}^{2}-10q+12=0$

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.

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.)

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.

$y=3{x}^{2}+6x+8$

up $x=-1$ $\left(-1,5\right)$ $y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,8\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none}$ minimum value of 5 when $x=-1$

$y={x}^{2}-4$

$y={x}^{2}+10x+24$

up $x=-5$ $\left(-5,-1\right)$ $y;\left(0,24\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-6,0\right),\left(-4,0\right)$ minimum value of $-5$ when $x=-1$

$y=-3{x}^{2}+12x-8$

$y=\text{−}{x}^{2}-8x+16$

down $x=-4$
$\left(-4,32\right)$ $y;\left(0,16\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-9.7,0\right),\left(1.7,0\right)$
maximum value of $32$ when $x=-4$

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

$y=2{x}^{2}+6x+2$

$y=16{x}^{2}+24x+9$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,9\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{3}{4},0\right);$
axis: $x=-\frac{3}{4};\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{3}{4},0\right)$

Solve.

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

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.
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
5%
Michael
hey everyone how to do algebra
Felecia answer 1.5 hours before he reaches her
I would like to solve the problem -6/2x
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year?
Tickets for the community fair cost $12 for adults and$5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold? Alpha Reply 220 gayla Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert? Tsimmuaj Reply Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj ? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year candido what would be wong candido 87 divided by 3 then multiply that by 4. 116 people total. Melissa the actual number that has 3 out of 4 of a whole pie candido was having a hard time finding Teddy use Matrices for the 2nd question Daniel One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers. Tsimmuaj Reply 26 + 37 = 63 + 8 = 71 gayla 26+37=63+8=71 ziad 11+52=63+8=71 Thisha how do we know the answer is correct? Thisha 23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for. gayla 23 +11 = 37. 23+37=63 63+8=71 Gayla by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71 Gayla your answer did not fit the guidelines of the question 11 is 41 less than 52. gayla 71-8-11 =52 is this correct? Ruel let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer tesfu just new Muwanga Amara currently sells televisions for company A at a salary of$17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of$29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal? Tsimmuaj Reply yes math Kenneth company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100 gayla need help with math to do tsi test Toocute me too Christian have you tried the TSI practice test ***tsipracticetest.com gayla DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players. gustavo Reply ? Ann DaMarcus: 16 mi/hr Fabian: 22 mi/hr Sherman Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution? Wenda Reply 15 and 5 32 is 40% , & 8 is 10 % , & any 4 letters is 5%. Karen It felt that something is missing on the question like: 40% of what solution? 10% of what solution? Jhea its confusing Sparcast 3% & 2% to complete the 25% Sparcast because she already has 20 liters. Sparcast ok I was a little confused I agree 15% & 5% Sparcast 8,2 Karen Jim and Debbie earned$7200. Debbie earned \$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike