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Algebraic

For the following exercises, solve the quadratic equation by factoring.

x 2 9 x + 18 = 0

x = 6 , x = 3

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6 x 2 + 17 x + 5 = 0

x = 5 2 , x = 1 3

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3 x 2 75 = 0

x = 5 , x = −5

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4 x 2 = 9

x = 3 2 , x = 3 2

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5 x 2 = 5 x + 30

x = −2 ,

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7 x 2 + 3 x = 0

x = 0 , x = 3 7

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For the following exercises, solve the quadratic equation by using the square root property.

x 2 = 36

x = −6 , x = 6

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( x 1 ) 2 = 25

x = 6 , x = −4

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( 2 x + 1 ) 2 = 9

x = 1 , x = −2

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For the following exercises, solve the quadratic equation by completing the square. Show each step.

x 2 9 x 22 = 0

x = −2 , x = 11

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x 2 6 x = 13

x = 3 ± 22

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2 + z = 6 z 2

z = 2 3 , z = 1 2

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2 x 2 3 x 1 = 0

x = 3 ± 17 4

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For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

x 2 + 4 x + 7 = 0

Not real

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9 x 2 30 x + 25 = 0

One rational

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6 x 2 x 2 = 0

Two real; rational

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For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution .

x 2 + x = 4

x = 1 ± 17 2

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3 x 2 5 x + 1 = 0

x = 5 ± 13 6

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4 + 1 x 1 x 2 = 0

x = 1 ± 17 8

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Technology

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x -intercepts) by using 2 nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

Y 1 = −3 x 2 + 8 x 1

x 0.131 and x 2.535

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To solve the quadratic equation x 2 + 5 x 7 = 4 , we can graph these two equations

Y 1 = x 2 + 5 x 7 Y 2 = 4

and find the points of intersection. Recall 2 nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

x 6.7 and x 1.7

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To solve the quadratic equation 0.3 x 2 + 2 x 4 = 2 , we can graph these two equations

Y 1 = 0.3 x 2 + 2 x 4 Y 2 = 2

and find the points of intersection. Recall 2 nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

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Extensions

Beginning with the general form of a quadratic equation, a x 2 + b x + c = 0 , solve for x by using the completing the square method, thus deriving the quadratic formula.

a x 2 + b x + c = 0 x 2 + b a x = c a x 2 + b a x + b 2 4 a 2 = c a + b 4 a 2 ( x + b 2 a ) 2 = b 2 4 a c 4 a 2 x + b 2 a = ± b 2 4 a c 4 a 2 x = b ± b 2 4 a c 2 a

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Show that the sum of the two solutions to the quadratic equation is b a .

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A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft. 2 . Solve the quadratic equation to find the length and width.

x ( x + 10 ) = 119 ; 7 ft. and 17 ft.

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Abercrombie and Fitch stock had a price given as P = 0.2 t 2 5.6 t + 50.2 , where t is the time in months from 1999 to 2001. ( t = 1 is January 1999). Find the two months in which the price of the stock was $30.

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Suppose that an equation is given p = −2 x 2 + 280 x 1000 , where x represents the number of items sold at an auction and p is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2 nd CALC maximum. To obtain a good window for the curve, set x [0,200] and y [0,10000].

maximum at x = 70

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Real-world applications

A formula for the normal systolic blood pressure for a man age A , measured in mmHg, is given as P = 0.006 A 2 0.02 A + 120. Find the age to the nearest year of a man whose normal blood pressure measures 125 mmHg.

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The cost function for a certain company is C = 60 x + 300 and the revenue is given by R = 100 x 0.5 x 2 . Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of x (production level) that will create a profit of $300.

The quadratic equation would be ( 100 x −0.5 x 2 ) ( 60 x + 300 ) = 300. The two values of x are 20 and 60.

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A falling object travels a distance given by the formula d = 5 t + 16 t 2 ft, where t is measured in seconds. How long will it take for the object to traveled 74 ft?

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A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of 378 ft 2 . Find the width of the walkway if the garden is 12 ft. wide by 15 ft. long.

A rectangle inside of a larger rectangle. The smaller rectangle has the length labeled: 15 feet and the width labeled: 12 feet. The distance between the two rectangles is labeled as x on all four sides.

3 feet

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An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, P , who contracted the flu t days after it broke out is given by the model P = t 2 + 13 t + 130 , where 1 t 6. Find the day that 160 students had the flu. Recall that the restriction on t is at most 6.

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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