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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

the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid 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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