solve by completing the square

The least polynomial with degree 3 is equal to f(x) = x³ - (8 / 7) · x² + 4 · x - 32 / 7.

In this problem we need to determine the least polynomial such that its degree is 3 and - i 2 and 8 / 7 are roots of the resulting expression. By quadratic formula, the roots - i 2 is along with its conjugate i 2 to obtain a quadratic equation with real coefficients. The factor form of the cubic polynomial is:

f(x) = a · (x - r₁) · (x - r₂) · (x - r₃)

Where:

And we must determine the standard form of the polynomial by means of algebra properties. This form is shown below:

f(x) = a · x³ + b · x² + c · x + d

Where a, b, c, d are real coefficients.

First, write the polynomial in factor form: (a = 1)

f(x) = (x - 8 / 7) · (x + i 2) · (x - i 2)

f(x) = (x - 8 / 7) · (x² + 4)

f(x) = x³ - (8 / 7) · x² + 4 · x - 32 / 7

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After the markup, the selling price of the item will be $39.

We know that a store owner decides to mark up all items by 30%, so basically we need to increase all the costs by 30%.

If the original cost is C, then the cost after the mark up will be:

P = C*(1 + 30%/100%)

P = C*(1 + 0.3)

P = C*(1.3)

In this case, we want to find the selling price P for an object whose original cost C is $30, then we can replace C in the equation above so we get:

P = $30*1.3 = $39

The selling price P  is $39.

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A mathematical expression made up of variables, coefficients, constants, and operations like addition, subtraction, multiplication, and division is known as an algebraic expression.

910x − 2

simplify | 910 x - 2

2 (455 x - 1).

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Ages of Tristan, Richard, and Pablo are in the order 6:7:4.

Detailed explanation:

Pablo is six years younger than Richard, seven years older than Tristan, and five years younger than Richard.

We can multiply the ratios of Pablo's age to Richard's age and Pablo's age to Tristan's age to determine the ratio of Tristan's age to Richard's age before determining the ratio of Tristan's age to Pablo's age.

These ratios add up to 6/7 * 5/4 = 30/28.

Pablo's age to Richard's age is 6:7, thus we may multiply this ratio by the age of Richard by Tristan's age to determine the age of Pablo, Richard, and Tristan.

Thus, the ratio of Pablo's age to Richard's age to Tristan's age is 6:7:4.

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Slope of Parallel line is 3/4

Slope of Perpendicular line is -4/3

What is Slope of Line?

The slope of a line indicates its steepness. Slope is computed mathematically as "rise over run" (change in y divided by change in x).

Solution:

General Form of Line: y = mx + c

here, m is the slope

4y = 3x + 6

y = 3/4x + 3/2

Parallel Line have the same Slope = 3/4

Perpendicular Line:

Product of Slope = -1

Slope of Perpendicualr Line = -4/3

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The standard form equation for the line through the points (3,-4) and (6,-2) is  -2/3 x + y = 4.

What is the equation of the line passing through the two-point?

To find the equation for the line through two points we can use the formula:

y - y1 = m(x - x1)

where (x1, y1) is one of the points, and m is the slope of the line. The slope of the line is equal to the change in y over the change in x, which can be calculated as:

m = (y2 - y1) / (x2 - x1)

where (x2, y2) is the other point.

Substituting the given points and the formula for the slope into the formula for the equation of the line, we get:

y - (-4) = ((-2) - (-4)) / (6 - 3) (x - 3)

This simplifies to:

y + 4 = -2/3 (x - 3)

To write the equation in standard form, we can distribute the -2/3 and then rearrange the terms:

-2/3 x + y = 4

Hence, the standard form equation for the line through the points (3,-4) and (6,-2) is  -2/3 x + y = 4

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Answer: no

Step-by-step explanation:

Recipe calls for a 3:4 ratio of raisins to peanuts

If his intention was to make a 1.5x recipe, he should have used 6 cups peanuts and 4-1/2 cups raisins (not 5!) to make the recipe in the correct ratio.

Answer: (x-2)(x-10)

Step-by-step explanation:

1. Factor out 3x by diving everything by it, this makes the monomial 3x(x^2-12x+20)

2. From here, you do your basic factoring and reach that the answer is (x-2)(x-10)

lmk if you need more details for explanation

Answer:

Step-by-step explanation:

to put into an equation:

8c=6.00

c=6.00/8

c=$0.75

The cost of one apple is $0.75

Answer:

25%

Step-by-step explanation:

The required probability of winning is given as 0.32.

Given that,
Playing a game that involves tossing a single die. If roll one, you win $100. If you roll a 2, you win $50. roll anything else, you lose $25.

Probability can be defined as the ratio of favorable outcomes to the total number of events.

here,
total number of sample spaces = 6
total of favorable outcome = 2
Probability fo the winning = 2 / 6 = 0.3333 or 0.32

Thus, the required probability of winning is given as 0.32.

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The correct option B. double; angle used to rotate the xy-coordinate plane to a new x'y'-plane for graphing a rotated conic section.

Any curve created by intersection of such a plane as well as a right circular cone is known in geometry as a conic section, sometimes known as a conic.

Thus, teta is a double angle that is used to rotate the x-y coordinate plane into a new x'y' plane in order to graph a rotated conic section.

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The ending Raw Materials Inventory balance for March is 31,800.

What is arithmetic?

Arithmetic is an elementary a part of arithmetic that consists of the study of the properties of the normal operations on numbers—addition, subtraction, multiplication, division, involution, and extraction of roots.

Main Body:

We use the Inventory identity to solve for ending inventory

Beginning Inventory + Purchase = ending Inventory + used

20,800 +159,000 = ending inventory + used materials

used materials:

135,600 direct

12,400 indirect

148,000 total materials used

20,800 + 159,000 = Ending Inventory + 148,000

20,800 + 159,000 - 148,000 = Ending Inventory

Ending Inventory = 31,800

Hence the ending Raw Materials Inventory balance for March is 31,800.

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Answer:

y = -2x + 5

Step-by-step explanation:

To find the slope use rise over run

To find the y-intercept see where the line intersects in the y-axis line

Answer:

2.4 and 2

Step-by-step explanation:

x-0.4

Answer:

2.4, 2, 1.6

Explanation:

The sequence is counting down by 0.4.

Answer:

$5000

Step-by-step explanation:

Let x represent the amount invested at 6% interest. Then 7000-x is the amount invested at 7% interest

6% interest = 6/100 = 0.06 in decimal

7% interest = 7/100 = 0.07

Interest earned from depositing x dollars at 6% = 0.06x

Interest earned from depositing 7000-x at 7% = 0.07(7000-x)

Total interest earned would be
0.06x + 0.07(7000-x)

Since this must be at least $440 we get the equation interest as

0.06x + 0.07(7000-x) = 440

Simplifying
0.06x + 0.07 x 7000 - 0.07x = 440

0.06x + 490 -0.07x = 440

-0.01x = 440 - 490

-0.01x = -50

Multiply both sides by -1

==> 0.01x = 50

==> x = 50/0.01 = 5000

So max amount that can be invested at 6% interest = $5000

Answer:

1/4 and -20

Step-by-step explanation:

"Slope" just means the change in y divided by the change in x. In the first graph, the slope is the change in depth divided by the change in hours.

Using the equation, slope = (y - y1)/(x - x1), you can plug in 2 points and solve. I see that the line goes through the points (4,1) and (8,2), so I'll use these. Make sure that (x,y) are from the same point and (x1,y1) are from the same point.

slope = (2 - 1)/(8 - 4)

slope = 1/4

You can do the same process for the second graph but you'll use different points. I see the points (2,80) and (4,40), so I'll use those.

slope = (40 - 80)/(4 - 2)

slope = -40/2

slope = -20

Answer:

x = 3

Step-by-step explanation:

Since ∠CAT = ∠DOG, ∠CAT = ∠DGO

Therefore AC = DO

8x - 6 = 3x + 9

8x - 3x = 9 + 6

5x = 15

x = 15/5

x = 3

The freight train is 88.88 feet long.

Speed = Distance/Time

Distance = Speed X Time

Time = Distance / Speed

As the trains travel in opposite direction then

Relative speed = (42 + 18) = 60 Km/ hr

                                            = 60 x 5/18 = 300/ 18

                                             = 100/9 m/s

So, length of the freight train = 100/9 x 8

                                               = 88.88 feet

Hence, The freight train is 88.88 feet long.

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Hello,

I hope you and your family are doing well!

a. The variable t represents the number of years since 1999, so the constraints for t can be written as t ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

b. To find the value of S(7), we substitute t = 7 into the function: S(7) = 1.084(7) + 17.392 = 22.688.

c. In context, the value of S(7) represents the amount of money, in billions of dollars, that the US government spent on science, space, and technology 7 years after 1999.

d. The key feature that is found when solving S(t) = 0 is the y-intercept of the graph of S(t). The y-intercept is the point where the graph crosses the y-axis.

e. The value for S(t) = 0 is not reasonable for the context because it represents the amount of money spent on science, space, and technology in 1999, which would be a negative number of years after 1999.

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Happy Holidays!

The incumbent is the current holder of an office or position, usually in relation to an election.

The variable of interest is:

X: Number of people that support the candidate.

You need to calculate the sample size to estimate the population proportion of supporters given a confidence level of 99% and a margin of error of 4%

The margin of error of the confidence level for the population proportion is

d = [tex]Z_{1} - \frac{\alpha }{2} * \sqrt{\frac{p^{|}(1 - p^{|} ) }{n} }[/tex]

=[tex]\frac{d}{Z_{1}-\alpha /2 } =\sqrt{\frac{p^{2}(1-p^{2} ) }{n} }[/tex]

sample proportion "p'" since there is no sample information, nor any previous information is known, you have to consider it as the "worse case scenario" and use the value of p'= 0.50

d= 0.04

She has to take a sample of 1045 people to estimate the population proportion of her supporters with a confidence level of 99% and a margin of error of 4%

=[tex]0.5*0.5*(\frac{2.586}{0.04} } )^{2} =1044.9=1045[/tex]

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Answer:

i think it would be a; When the price of housing increases, more homeowners want to sell their house.

The tables that represent a function are:

Input x     5 8 −8 −3 2

Output y 12 17 10 5 2

Input x     18 20 22 24 26

Output y  -9 -9  -6  -3   0

Input x  −4 8 3 −12 0

Output y 13 13 13 13 13

Options A, B, and D are the correct answer.

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

A function can have many-to-one but not one-to-many.

Now,

The tables are given as,

Input x     5 8 −8 −3 2

Output y 12 17 10 5 2

This is a function.

Input x     18 20 22 24 26

Output y  -9 -9  -6  -3   0

This is a function.

Input x    24 32 45 12 32

Output y 11 12 35 14 35

This is not a function because it has one-to-many.

Input x  −4 8 3 −12 0

Output y 13 13 13 13 13

This is a function.

Input x    25 30 30 25 20

Output y  6 −2 0 13 11

This is not a function because it has one-to-many.

Thus,

The tables that represent a function are:

Input x     5 8 −8 −3 2

Output y 12 17 10 5 2

Input x     18 20 22 24 26

Output y  -9 -9  -6  -3   0

Input x  −4 8 3 −12 0

Output y 13 13 13 13 13

Options A, B, and D are the correct answer.

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Answer:

function 2

Step-by-step explanation:

function 1 has a slope of 8/7 which is equal to 0.75 and function 2 has a slope of 1 so function 2 is greater than function 1.

Answer: <

Step-by-step explanation:

One number is less than another number if their difference is less than zero.

Since x=2y-4 - 0.75 = -0.75 which is less than zero, x=2y-4 is less than 0.75

Therefore, x=2y-4 is 0.75 less/lower than 0.75

Answer:

x = 55

Step-by-step explanation:

[tex]\frac{x-5}{10}[/tex] = 5 ( multiply both sides by 10 to clear the fraction )

x - 5 = 10 × 5 = 50 ( add 5 to both sides )

x = 55

Answer:

Step-by-step explanation:

Find the equation of lines m, n and p

                    The equation of line:      

                       [tex]\boxed {\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1} }[/tex]

Line m: (-3,2)  (3,6)

x₁=-3   x₂=3   y₁=2   y₂=6

[tex]\displaystyle\\\frac{x-(-3)}{3-(-3)} =\frac{y-2}{6-2} \\\\\frac{x+3}{3+3} =\frac{y-2}{4} \\\\\frac{x+3}{6}=\frac{y-2}{4}[/tex]

Multiply both parts of the equation by 4:

[tex]\displaystyle\\\frac{x+3}{6}(4)=y-2\\\\\frac{2}{3} (x+3)=y-2\\\\\frac{2}{3}x+2=y-2\\\\\frac{2}{3}x+2+2=y-2+2\\\\\frac{2}{3}x+4=y\\\\Thus,\ y=\frac{2}{3} x+4\\\\Hence,\ m_m=\frac{2}{3}[/tex]

Line n: (-5,-5)  (5,1)

x₁=-5   x₂=5   y₁=-5   y₂=1

[tex]\displaystyle\\\frac{x-(-5)}{5-(-5)}=\frac{y-(-5)}{1-(-5)} \\\\\frac{x+5}{5+5}=\frac{y+5}{1+5} \\\\\frac{x+6}{10}=\frac{y+5}{6}[/tex]

Multiply both parts of the equation by 6:

[tex]\displaystyle\\\frac{3}{5}( x+5)=y+5\\\\\frac{3}{5} x+3 =y+5\\\\\frac{3}{5} x+3-5=y+5-5\\\\\frac{3}{5} x-2=y\\\\Thus,\ y=\frac{3}{5}x-2\\\\Hence,\ m_n=\frac{3}{5}[/tex]

Line p: (3,-4)  (-3,6)

x₁=3   x₂=-3   y₁=-4   y₂=6

[tex]\displaystyle\\\frac{x-3}{-3-3} =\frac{y-6}{6-(-4)} \\\\\frac{x-3}{-6} =\frac{y-6}{6+4} \\\\\frac{x-3}{-6}=\frac{y-6}{10} \\[/tex]

Multiply both parts of the equation by 10:

[tex]\displaystyle\\-\frac{5}{3} x+5=y-6\\\\-\frac{5}{3} x+5+6=y-6+6\\\\-\frac{5}{3} x+11=y\\Thus,\ y=-\frac{5}{3}x+11 \\\\Hence,\ m_p=-\frac{5}{3}[/tex]

[tex]Question# 5: \ m_m\neq m_n\ \ \ \ m\ isn't\ parallel \ to\ n\\\\Question# 6:\ m\ isn't \ congruent \ to\ p\\\\Question#7 :\ n\ isn't \ congruent \ to\ p[/tex]

[tex]n \bot p[/tex]

Answer:

5) No the lines are not parallel because in order for this to be true, the slopes have to be equal. Line m has a slope of 2/3 and line n has a slope of 3/5. Both slopes are not equal so it is not parallel

6) No, m is not congruent to p

7) No, n is not congruent to p

(Is there any chance that you meant to say perpendicular instead of "congruent"?)

Step-by-step explanation:

For two lines to be parallel the slopes have to be equal

See attached to see the rise over run

line m: 2/3

line n: 3/5

The slopes are not equal so the lines are not parallel.

Answer:

9

Step-by-step explanation:

[tex]\frac{12}{4}+3(2) \\ \\ =3+6 \\ \\ =9[/tex]

Answer:

Answer is 9.

Step-by-step explanation:

If x is 12, then do 12 divided by 4, and you get 3. If y is 2, then di 3 times 2, which is 6. 3 plus 6 is 9.

x=6

d=12

7x+12 or 7x+d

7(6) + 12 or (7x6) + 12 = 54

78-12 = 66

66/6 = 11

11x + 12 =78.

Answer:

Problem

Solution

Solve for the missing components of the triangle.

A

=

69.62591065

B

=

65.69947977

C

=

44.67460956

a

=

18

b

=

17.5