solve polynomial equations by factoring

2x³ + 12x² = 0

By using distance formula the following results are obtained

a) Other value of k = 14

b) For k = 4, the equation of bisector of [tex]\angle ABC[/tex] is

[tex]y = \frac{1}{3}x + 4[/tex]

What is distance formula?

Distance formula is used to find the distance between two coordinates.

If the coordinates of one point be ([tex]x_1, y_1[/tex]) and coordinates of other point is  ([tex]x_2, y_2[/tex]), then  distance between the two points is

[tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2[/tex]

The points A, B and C have coordinates A(2, 8), B(9, 7) and C(k, k-2).

Length of AB =

[tex]\sqrt{(9 - 2)^2 + (7 - 8)^2}\\\\\sqrt{49 +1}\\\\\sqrt{50}[/tex]

Length of BC =

[tex]\sqrt{(k - 9)^2 + (k - 2 - 7)^2}\\\\\sqrt{(k - 9)^2 + (k - 9)^2}\\\\\sqrt{2(k-9)^2}\\\\(k-9) \sqrt{2}[/tex]

By the problem,

[tex](k-9)\sqrt{2} = \sqrt{50}\\k - 9 = \sqrt\frac{50}{2}}\\k - 9 = \sqrt{25}\\k - 9 = \pm 5\\k - 9 = 5 \ or \ k - 9 = -5\\k = 5 + 9 \ or \ k = 9 -5\\k = 14 \ or \ k = 4[/tex]

The other value of k is 14

b) When  k = 4

Coordinate of C = (4, 2)

Since AB =AC , the bisector of [tex]\angle ABC[/tex] passes through the midpoint of AC

Midpoint of AC =

[tex](\frac{2 + 4}{2} , \frac{8+2}{2})\\(3, 5)[/tex]

The line also passes through (9, 7)

Slope of the  bisector of [tex]\angle ABC[/tex] = [tex]\frac{7 -5}{9 - 3}[/tex]

                                                    = [tex]\frac{1}{3}[/tex]

Equation of the bisector of [tex]\angle ABC[/tex]

[tex]y - 5 = \frac{1}{3}(x - 3)\\\\y - 5 = \frac{1}{3}x -1\\\\y = \frac{1}{3} x - 1 + 5\\\\y = \frac{1}{3} x + 4[/tex]

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The initial value of f(x) is -36.

The initial value of g(x) is -44

The range of f(x) is {y | -36 ≤ y ≤ -20}

The range of g(x) is {y | -44 ≤ y ≤ -20}

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,

{x | -6 ≤ x ≤  -2}

g(x) = 6x - 8

From the table, we get that,

The initial value of f(x) is -36.

The initial value of g(x).

g(x) = 6 x (-6) - 8 = -36 - 8 = -44

The range of f(x).

= {y | -36 ≤ y ≤ -20}

g(x) = 6x - 8

g(x) = 6 x (-6) - 8 = -36 - 8 = -44

g(x) = 6 x (-2) - 8 = -12 - 8 = -20

The range of g(x)

= {y | -44 ≤ y ≤ -20}

Thus,

The initial value of f(x) is -36.

The initial value of g(x) is -44

The range of f(x) is {y | -36 ≤ y ≤ -20}

The range of g(x) is {y | -44 ≤ y ≤ -20}

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The distance between (7,-2) and (-2,3) is 11.2 units using the distance formula.

Distance is a measurement of how far apart two objects or points are, either numerically or occasionally qualitatively. Distance can refer to a physical length in physics or to an estimate based on other factors in common usage. a measurement of the distance between two points along a line or line segment. It's formula is "distance formula=√(x2-x1)²+(y2-y1)²".

Here,

distance formula=√(x2-x1)²+(y2-y1)²

=√(-2-7)²+(3--2)²

=√(-9)²+5²

=√81+45

=√126

=11.225 units

rounding to nearest tenth,

=11.2 units

Using the distance formula, the distance between (7,-2) and (-2,3) is 11.2 units.

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Answer:

Step-by-step explanation:

hope it helps

Answer:

105

Step-by-step explanation:

1. x = mushroom he gather

2. he dry 2/5 from he gather, 2/5 from x

3. he pickled 5/9 of remaining mushroom

find the remaining mushroom value first, x-2/5x = 3/5x

then multiply with 5/9, 5/9(3/5x) = 15/45x

4. total mushroom - dry - pickled, he still have 28 to fry

5. write the equation like this

x - 2/5x - 15/45x = 28

45/45x - 18/45x -15/45x = 28

12/45x = 28

x = 28(45) / 12

x = 105

On October 24, 218 days after March 20, there will be 10 hours of sunlight.

The number of daily sunlight hours of a particular location is modelled by the trigonometric equation,

l(t) = 12 + 3.5sin (2[tex]\pi[/tex]/365t)

Here , L(t)= Number of sunlight hours in a day.

           t = Number of days after 20th March .

Now we are required to determime the day  during first 365 on which there are 10 sunlight hours ,that is, we need the value of "t" for which L(t) =10 hours. So in the given equation by putting L(t)= 10 and solving for t , follows as,

l(t) = 12 + 3.5sin (2[tex]\pi[/tex]/365t)

10 = 12 + 3.5sin (2[tex]\pi[/tex]/365t)

10-12 = 3.5sin (2[tex]\pi[/tex]/365t)

-2 =  3.5sin (2[tex]\pi[/tex]/365t)

-2/3.5 = sin (2[tex]\pi[/tex]/365t)

-0.574 = sin (2[tex]\pi[/tex]/365t)

For the inverse function of y = sin x

sin ⁻¹(y) =x + 2n[tex]\pi[/tex]

Now as we are required to calculate the day having 10 sunlight hours in first 365 days so putting n=0 we get,

t = -35.3521 or t =217.9422

Now t can not be negative as we want to find the day after march 20 having 10 sunlight hours. So neglecting t=-35.3521 . Now we get t=217.9422 .

Rounding off we get ,t=218 which is the required answer.

Hence, 218 days after march 20 i.e on October 24 it will have 10 sunlights.

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The amount of money Rita will get after 5 year with interest rate compounds monthly is $4,420.

The formula for the calculation of the compound interest is-

A = P(1 + r/100n)∧nt

In which,

A  = amount after compounding

P = principal amount (  $20,000 )

n = number of time principal amount compounded in a year. ( 12 )

t = total time in years. (5 )

r = rate of interest (4%)

Put the value in eq.

A = 20,000(1 + 4/1200)∧5×12

A = 20,000×1.22

A = 24420

CI = A - P

CI = 24420 - 20000

CI = 4,420.

Thus, the amount of money Rita will get after 5 year with interest

rate compounds monthly is $4,420.

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Answer:

[tex]-12x+6[/tex]

Step-by-step explanation:

Take a look at the attachment,,,

Hope this helps! :)

Answer:

[tex]6 + 12x[/tex]

Step-by-step explanation:

The distributive property states that A(B + C) = AB + AC.

Therefore,

[tex]4 \left( \dfrac{3}{2} - 3x \right) = 4 \left( \dfrac{3}{2} \right) + 4(3x)[/tex]

Now we can simplify each term:

[tex]4 \left( \dfrac{3}{2} \right) + 4(3x)[/tex]

[tex]= \dfrac{12}{2} + 12x[/tex]

[tex]= 6 + 12x[/tex]

The angle 1 is 88 degree, 2 is 42 degree and the 3 is 113 degree.

We know that the sum of the angles of the triangle is 180, so in one triangle where the 3 angle is missing we can write as:

42 + 25 +  ∠3 = 180

∠3 + 67 = 180 ,

∠3 = 113°

now in the triangle where 1 and 2 angles are missing, we can say that ∠2 = 42° because of transversely equal angles:

so, ∠ 2 = 42°

now again for a triangle,

∠1 + 42 + 50 = 180

∠1 + 92 = 180

∠1 = 88°

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The volume of gas consumed by the first and second cars is 25 gallons and 30 gallons, respectively.

A family has two cars. The first car has a fuel efficiency of 25 miles per gallon of gas, and the second has a fuel efficiency of 30 miles per gallon of gas. During one particular week, the two cars travelled a combined total of 1525 miles, for a total gas consumption of 55 gallons.

Let the volumes of gas consumed by the first car and the second car be denoted by the variables "x" and "y". We can form the given two equations using the given data.

x + y = 55

25x + 30y = 1525

We will substitute the value of "x" from the first equation into the second equation.

x = 55 - y

25x + 30y = 1525

25(55 - y) + 30y = 1525

1375 - 25y + 30y = 1525

5y = 150

y = 30

x = 55 - y = 55 - 30 = 25

Hence, the gas consumed by the first and second cars is 25 gallons and 30 gallons, respectively.

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Answer:

It will take 8 days

Step-by-step explanation:

Mixed numbers in equations

1 3/4 * d = 14

Change the mixed number to an improper fraction

1 3/4 = ( 4*1 + 3) /4 = 7/4

7/4 * d = 14

Multiply each side by 4/7

4/7 * 7/4  *d = 14 * 4/7

d = 8

It will take 8 days.

Each worker looked 40 passengers

From the question, we have

On a plane there were 6 airline workers and 240 passengers.

each worker looked after the same number of passengers

number of passengers = 240/6

=40

Each worker looked 40 passengers

Divide:

Repetitive subtraction is the process of division. It is the multiplication operation's opposite. It is described as the process of creating equitable groups. When dividing numbers, we divide a larger number down into smaller ones such that the larger number obtained will be equal to the multiplication of the smaller numbers. One of the four fundamental mathematical operations, along with addition, subtraction, and multiplication, is division. Division is the process of dividing a larger group into smaller groups so that each group contains an equal number of things. It is a mathematical operation used for equal distribution and equal grouping.

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Answer:

10 72

Step-by-step explanation:

The difference of the polynomials [tex]2x^2[/tex] - 9x + 8 and [tex]6x^2[/tex] - 5x - 12 is -[tex]4x^2[/tex] - 4x 20

The first polynomial is given that

[tex]2x^2[/tex] - 9x + 8

The second polynomial is

[tex]6x^2[/tex] - 5x - 12

Here we have to find the difference, to find that we need to subtract the second polynomial from the second polynomial

Then,
[tex]2x^2[/tex] - 9x + 8 - ( [tex]6x^2[/tex] - 5x - 12 )

Open the bracket using the distributive property

[tex]2x^2[/tex] - 9x + 8 - [tex]6x^2[/tex] + 5x +12

Combine the like terms

[tex]2x^2[/tex] -  [tex]6x^2[/tex]  -9x +5x + 8 +12

Apply the arithmetic operations on like terms

-[tex]4x^2[/tex] - 4x 20

Hence, the difference of the polynomials [tex]2x^2[/tex] - 9x + 8 and [tex]6x^2[/tex] - 5x - 12  is   -[tex]4x^2[/tex] - 4x 20

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The equation of line v is y - x = 7 and The equation in slope-intercept form is y = x + 7.

Given:

y = -x + 7

Compare it to Slope-intercept form y = mx + c (∵ m = slope or gradient)

The gradient (m₁) is -1.

The gradient of the perpendicular line v is given by

m₂= -1/m₁

m₂=-1/-1

m₂=1

To find the duration of the perpendicular line v use the formula

y- y₁ = m(x-x₁)   (i.e., one slope, one point formula)

y₁=-1, x₁ =-8 , m₂ =1

y-(-1) = 1[x-(-8)]

y + 1 = x + 8

y - x = 8 - 1

y - x = 7

So, The equation of line v is y - x = 7 and The equation in slope-intercept form is y = x + 7.

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The equation for the line of best fit for a woman's height, y, based on her shoe size, x is y = 1.36x + 55.6.

From the given table:

To find the equation we need to insert the points (x,y) in calculator.

Points are:

Sum of X = 91

Sum of Y = 624.5

Mean X = 10.1111

Mean Y = 69.3889

Sum of squares (SSX) = 42.8889

Sum of products (SP) = 57.6111

Regression Equation = ŷ = bX + a

b = SP/SSX = 57.61/42.89 = 1.34326

a = MY - bMX = 69.39 - (1.34*10.11) = 55.80699

y = 1.34326X + 55.80699

aproximate:

y = 1.36 + 55.6.

Therefore The equation for the line of best fit for a woman's height, y, based on her shoe size, x is y = 1.36x + 55.6.

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the answer for your question above is (-1,-1) or you could say option no.3

For the given quadratic equation:

y = x² - 6x - 7

The vertex is (3, -16) and the axis of symmetry is at x = 3.

For a quadratic equation of the form:

y = a*x² + b*x + c

The x-value of the vertex (h, k) is:

h = -b/2a

And the axis of symmetry is the vertical line x = h.

In this case, the quadratic equation is:

y = x² - 6x - 7

The x-value of the vertex is at:

h = -(-6)/2*1 = 3

And the y-value of the vertex is at:

y = (3)² - 6*3 - 7

y = 9 - 18 - 7

y = -16

Then the vertex is at (3, -16)

And the axis of symmetry is x = 3

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a)An expression for the value of the account at the end of three years will be:

$1000[tex]x^3[/tex] +300[tex]x^2[/tex] + 500x

b) The amount in the account at the end of three years if the interest rate is 4% will be:

= 1000 × [tex]1.04^3[/tex] + 300 × [tex]1.04^2[/tex] + 500 × 1.04

= $ 1969.344

Given,

In the question:

A growth factor of x = 1+r. $1,000 is deposited at the start of the first year,

An additional $300 is deposited at the start of the next year, and $500 at the start of the following year.

To find the:

a) Write an expression for the value of the account at the end of three years in terms of the growth factor x.

b) Determine (to the nearest cent) the amount in the account at the end of three years if the interest rate is 4%.

Now, According to the question:

a) Growth factor x = 1 + r

Based on the given conditions:

An expression for the value of the account at the end of three years will be:

$1000[tex]x^3[/tex] +300[tex]x^2[/tex] + 500x

b) The amount in the account at the end of three years if the interest rate is 4% will be:

1 + 4% = 1 + 0.04 = 1.04

= 1000 × [tex]1.04^3[/tex] + 300 × [tex]1.04^2[/tex] + 500 × 1.04

= $ 1969.344

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(a) she land to minimize the total travel time is 2.4min

(b) the minimum speed at which she must row so that the quickest way to the restaurant is to row directly is 6

(a)

T(x) = time rowing + time walking

     = [tex]\frac{\sqrt{x^{2} +16} }{2}[/tex] +[tex]\frac{6 - x}{3}[/tex]

T(x) = 1/2[tex]\sqrt{(x^{2} + 16)}[/tex]  + 2 - [tex]\frac{x}{3}[/tex]

  first derivation

T'(x) = 1/4[tex]\sqrt[-1/2]{(x^{2} + 16)}[/tex] .2x - 1/3

       = [tex]\frac{x}{2\sqrt{(x^{2} + 16)} }[/tex] - 1/3

T'(x) = 3x - 2 [tex]\sqrt{(x^{2} + 16)}[/tex] / 6[tex]\sqrt{(x^{2} + 16)}[/tex]

T'(x) = 0

3x - 2 [tex]\sqrt{(x^{2} + 16)}[/tex] / 6[tex]\sqrt{(x^{2} + 16)}[/tex]  = 0

3x - 2 [tex]\sqrt{(x^{2} + 16)}[/tex] =0

3x/2 = [tex]\sqrt{(x^{2} + 16)}[/tex]

9[tex]x^{2}[/tex]/4 = [tex]x^{2}[/tex] + 16

9[tex]x^{2}[/tex] = 4 [tex]x^{2}[/tex] + 64

5 [tex]x^{2}[/tex] = 64

[tex]x^{2}[/tex] =64/5

x = [tex]\sqrt{64/5}[/tex]

x = 8/[tex]\sqrt{5}[/tex]

x = 8[tex]\sqrt{5}[/tex]/5

total travel time = 6 - 8[tex]\sqrt{5}[/tex]/5 = 2.4 min

she land to minimize the total travel time is 2.4min

(b) T(x) = time rowing

           = [tex]\frac{\sqrt{x^{2} +16} }{2}[/tex]

first derivation

T'(x) = 1/4[tex]\sqrt[-1/2]{(x^{2} + 16)}[/tex] .2x

      = [tex]\frac{x}{2\sqrt{(x^{2} + 16)} }[/tex]

T'(x) = 0

[tex]\frac{x}{2\sqrt{(x^{2} + 16)} }[/tex] =0

x = 0

minimum speed = 6-x = 6-0=6

the minimum speed at which she must row so that the quickest way to the restaurant is to row directly is 6

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Answer:

positive

Step-by-step explanation:

Answer:

yes

Step-by-step explanation:

every x value has at least one y value

Answer:

Hence, he booked 19 tickets for Airlines A, 12 tickets for Airline B, and 41 tickets for Airlines C.

Step-by-step explanation:

Suppose the office manager booked a+b+c = 72 tickets.

a = b + 7

c = 3b + 5.

So, we have (b + 7) + b +(3b + 5) = 72, or

5b + 12 = 72, or

5b = 60, or

b = 12.

The expansion of the series  ⁴∑ₙ = ₁ 3ⁿ ⁺ ¹ is; 9 + 27 + 81 + 243

We are given the summation formula of the series as;

⁴∑ₙ = ₁ 3ⁿ ⁺ ¹

At n = 1, we will input 1 for n and we have;

3¹ ⁺ ¹ = 3² = 9

At n = 2, we will input 2 for n and we have;

3² ⁺ ¹ = 3³ = 27

At n = 3, we will input 3 for n to get;

3³ ⁺ ¹ = 3⁴ = 81

At n = 4;

3⁴ ⁺ ¹ = 3⁵ = 243

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The fraction left for Mark to read is 17/20.

From the information illustrated, he wants to complete of the 160 pages book by the end of the week and Mark has read 24 pages so far.

The number of pages left will be:

= 160 - 24

= 136 pages

Therefore, the fraction of pages left will be:

= Number of pages left / Total pages

= 136 / 160

= 17/20

The fraction is 17/20.

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The median for a particular set is one less than the mean.

Given;

List of order for goals scored by a soccer team in 9 matches

= 0, 0, 1, 1, 1, 3, 3, 4, 5

To find the relation between mean and median;

The obtained mean = Sum of observations/number of observations

                                 = 0 + 0 + 1 + 1 + 1 + 3 + 3 + 4 + 5

                                 = 18/9

                                 = 2

Mean = 2

For median;

let us write the given observations in ascending order;

⇒  0, 0, 1, 1, 1, 3, 3, 4, 5

Here no. of observations is r = 9 which is an odd number.

Therefore observation at 5th is a middle position which has 1 as a value.

Median = 1

Therefore, the median is less than the mean by 1.

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The values of x and y which are the missing terms of the given triangle have their values as; x = ⁵/₂ and y = ⁹/₄

From the given triangle, we can see that the transverse line intersects both side lengths at their midpoints. Thus, we can say that;

3x - 4 = x + 1   ------(1)

⁴/₃y + 2 = 4y - 4   ------(2)

Let us solve equation 1 by rearranging to get;

3x - x = 1 + 4

2x = 5

x = ⁵/₂

Similarly, we can say that for equation 2;

4y - ⁴/₃y =  2 + 4

4y - ⁴/₃y = 6

Multiply through by 3 to get;

12y - 4y = 18

8y = 18

y = 18/8

y = ⁹/₄

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The object moves at a speed of 25.352 m/s

The subject of a formula is the variable that is being worked out. It can be recognized as the letter on its own on one side of the equals sign. It is a variable which is expressed in terms of other variables involved in the formula.

Formulas are written so that a single variable, the subject of the formula is on the L.H.S. of the equation. Everything else goes on the right side of the equation. We evaluate the formula by substituting for the literal numbers on the right hand side.

In the question given, we have an equation;

s = d/t

d = 180m, t = 7.1s

s = 180 / 7.1

s = 25.352 m/s

The speed (s) of the object is 25.352 m/s

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Answer:

7.07

Step-by-step explanation:

Draw a horizonal line and a vertical line to creat a right triangle.  The horizontal measurement would be 5 and so would the horizontal measurement.

Use the Pythagorean theorem

[tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex]  The c side is the length that you are looking for

[tex]5^{2}[/tex] + [tex]5^{2}[/tex] = [tex]c^{2}[/tex]

25 + 25 = [tex]c^{2}[/tex]

50 = [tex]c^{2}[/tex]

[tex]\sqrt{50}[/tex] = [tex]\sqrt{c^{2} }[/tex]

7.07 = c  Rounded to the nearest hundredths