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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

How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
Practice Key Terms 7

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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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