Solve the following using quadratic equations using the Zero Product Property. Show all steps neatly.
5x²+6x+2=-x+8
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The expression i⁵ when evaluated is √-1 or i

From the question, we have the following parameters that can be used in our computation:

i⁵

The above expression is a complex numbers expression

As a general rule, complex numbers are numbers that consist of both a real part and an imaginary part.

A complex number can be written in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Using the above as a guide, we have the following:

i⁵ = i⁴ * i

So, we have

i⁵ = (√-1)⁴ * i

Evaluate the exponent

i⁵ = 1 * i

So, we have

i⁵ = i or i⁵ = √-1

Hence, the solution to the expression i⁵ is √-1 or i

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The answer is: x^(2) + 20/(x-2)

Synthetic division is a method of dividing a polynomial by a linear factor. It is a shorthand version of long division that is quicker and easier to do.

Here are the steps to solve this problem using synthetic division:

1. Write the coefficients of the dividend in a row: 1 -2 0 10
2. Write the constant term of the divisor to the left of the row: 2 | 1 -2 0 10
3. Bring down the first coefficient: 2 | 1 -2 0 10
      |_______
       1
4. Multiply the first coefficient by the divisor and write the result under the second coefficient: 2 | 1 -2 0 10
      |___2___
       1  0
5. Add the second coefficient and the result: 2 | 1 -2 0 10
      |___2___
       1  0
6. Repeat the process with the remaining coefficients: 2 | 1 -2 0 10
      |___2__0_20
       1  0 0 20
7. The final row of numbers represents the coefficients of the quotient and the remainder: (x^(2) + 0x + 0) + 20/(x-2)

So, the answer is: x^(2) + 20/(x-2)

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We can conclude that -

The exponential function is of the form -

f(x) = eˣ

Given is to write the exponential growth and exponential decay equation.

{ 1 } -

The general exponential growth equation is -

[tex]$f(x)=a(1+r)^{x}[/tex]

{ 2 } -

The general exponential decay equation is -

[tex]$\frac {dN}{dt}= -\lambda N[/tex]

Option {A} and option {E} represent the exponential decay. Option {B}, {E} and {D} represent exponential growth.

Therefore, we can conclude that -

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Answer:

990 applicants

Step-by-step explanation:

We know

A university law school accepts 4 out of every 11 applicants.

If the school accepted 360 ​students, find how many applications they received.

To get from 4 to 360, we time 90

We take 11 times 90 = 990 applicants

So, they received 990 applicants.

The answer choice which represents the solution to the inequality given as required is; b > -90.

Quotient refers to a number resulting from the division of one number by another.

As evident in the task content; the given inequality is;

Eight more than the quotient of a number b and 45 is greater than 6.

The symbols of inequalities are;

Greater than >

Less than <

Equal to =

Greater than or equal to ≥

Less than or equal to ≤

Therefore, the algebraic expression is;

b/45 + 8 > 6

Therefore; b/45 > -2

b > -90.

Ultimately, the required inequality is; b/45 + 8 > 6 and it's solution is; b > -90.

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Inverse of the given matrix is

A⁻¹  =     [0.5   0.5 -0.5]
            [ 0    -0.5  -0.5]
            [-0.5 -0.5  -0.5]

the matrix in question is

−1 0 1

2 2 0

1 0 1

now, we are given that inverse of the matrix exists.
A⁻¹ = adjoint of A/|A|

The determinant of the matrix A is:
det(A) = (-1) [(2)(1) - (0)(0)] - 0 [(2)(1)-(0)(1)] + 1 [(2)(0) - (2)(1)] = -4

The adjoint of the matrix A is:

2 2 −2

0 −2 −2

−2 −2 −2

Therefore, the inverse of the matrix A is:

A⁻¹ =  1/(-4)  [2 2 −2]

                  [0 −2 −2]

                  [−2 −2 −2]

A⁻¹  =     [0.5   0.5 -0.5]
            [ 0    -0.5  -0.5]
            [-0.5 -0.5  -0.5]

So, the inverse of the matrix A is:

A⁻¹  =     [0.5   0.5 -0.5]
            [ 0    -0.5  -0.5]
            [-0.5 -0.5  -0.5]

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The answer of  lines intersect at the point is (1, 2, -1)

To determine whether the lines L1(t) = ⟨3 − t, 8 − 3t, 1 − t⟩ and L2(s) = ⟨−1 − 2s, 3 + s, s⟩ intersect, we need to find a value of t and s that make the x, y, and z components of both lines equal.

First, let's set the x components of both lines equal to each other:

3 − t = −1 − 2s

Next, let's set the y components of both lines equal to each other:

8 − 3t = 3 + s

Finally, let's set the z components of both lines equal to each other:

1 − t = s

Now we can solve for t and s. From the first equation, we get:

t = 4 + 2s

Substituting this value of t into the second equation, we get:

8 − 3(4 + 2s) = 3 + s

Simplifying this equation gives us:

-4 − 6s = 3 + s

Solving for s gives us:

s = -1

Substituting this value of s back into the first equation gives us:

t = 4 + 2(-1) = 2

So the lines intersect at t = 2 and s = -1. To find the point of intersection, we can substitute these values back into either line equation. Substituting into L1(t) gives us:

L1(2) = ⟨3 − 2, 8 − 3(2), 1 − 2⟩ = ⟨1, 2, -1⟩

Therefore, the lines intersect at the point (1, 2, -1).

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Answer:

3/5

Step-by-step explanation:

In your problem you have a right triangle.

For this problem, you need to know what sine is. Sin is [tex]\\\frac{opposite}{hypotenuse}[/tex]

In this problem we know that the hypotenuse is 20, but we do not know the opposite. The opposite refers to the side that is opposite to the angle. In this case, the angle we are using is c. Also, in this case, the name of the opposite side is ED. To figure out that side, we need to use the pythagorean theorem.

[tex]a^{2} +b^{2} =c^{2}[/tex]

A is 16, c is 20

Therfore we can plug those values into our formula

[tex]16^{2} +b^{2}=20^{2}[/tex]

Simplify:

[tex]256 + b^{2} =400[/tex]

Simplify even more:

[tex]b^{2} = 144[/tex]

Square root both sides:

[tex]b = 12[/tex]

Now we Know all 3 sides of our triangle: 20, 16, and 12.

Sin refers to OPPOSITE/HYPOTENUSE

Our opposite is 12, and our hypotenuse is 20.

Plugging those values into our formula gives us:

12/20

Simplify:

3/5.

THERFORE, THE ANSWER IS 3/5

**Also, this whole process seems long, but in reality it is very short and you'll master it in no time!

If the price of each hot dog is set at 0.8, the revenue to be obtained is $120.

Income is the amount of money or value that a person, family, company or country receives during a given period of time, whether in the form of salary, rents, profits, interest, dividends or other sources.

According to the formula R = -50P +1 60, we can find the expected revenue by plugging in the given price and solving for R.

R = -50P+160

R = -50(0.8) + 160

R = -40 + 160

R = 120

Therefore, the expected revenue generated by setting the price to be 0.8 is $120.

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The state space is {0, 1, 2, 3},

The transition matrix is : P = [[(1-p)³, 3p(1-p², 3p²(1-p), p^3], [(1-p)², 2p(1-p)²+2p(1-p)², 2p²(1-p)+2p²(1-p), 2p³], [(1-p), p(1-p)²+2p(1-p), p²(1-p)+2p², p³], [0, p(1-p), 2p², 3p³]],  the probability of one idle machine 3 nights later is 0.33554432.

The state space of (Xn : n ≥ 0) is the set of all possible states that the system can be in at the end of a given day. The state space is {0, 1, 2, 3}, where 0 corresponds to all machines being broken, 1 corresponds to one machine working and two broken, 2 corresponds to two machines working and one broken, and 3 corresponds to all machines working.

The transition matrix is given by:
P = [[(1-p)³, 3p(1-p², 3p²(1-p), p^3], [(1-p)², 2p(1-p)²+2p(1-p)², 2p²(1-p)+2p²(1-p), 2p³], [(1-p), p(1-p)²+2p(1-p), p²(1-p)+2p², p³], [0, p(1-p), 2p², 3p³]]

Given that 2 machines are idle tonight (i.e., Xn = 1), the probability of one idle machine 3 nights later (i.e., Xn+3 = 2) is given by the (1, 2) entry of P^3, where P^3 is the matrix obtained by multiplying P by itself 3 times.

Using the given value of p = 0.8, we can calculate P^3 and find the (1, 2) entry to obtain the desired probability. The (1, 2) entry of P^3 is 0.33554432, so the probability of one idle machine 3 nights later is 0.33554432.

Therefore, the answer to part c is 0.335544.

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Answer: [tex]y=(x-4)^2+(y+2)^2[/tex]

Step-by-step explanation:

Using the formula [tex]y=(x-h)^2+(y-k)^2[/tex] where the center is (h,k), find the center point. In this case, the center point is (4, -2). Plug it into the equation

The length of line BC is 10cm ,The area of the circle is 78.5 cm² and The circumstance of Circle  is  31.4 cm.  

Part-1

To Find the length of BC, We have the lenghts of CD=8cm and DE=12cm

To find out length. We have to make a perpendicular line construction joins AE.

Triangle AED forms a right angled triangle.

As we all know pythagoras theorem.

Pythagoras formula c²=a²+b²

To findout radius of circle

Let AE=AC=r  

      AD=CD+AC

      AD=8+r

                               AD²=AE²+DE²

Submit all values in the above equation.

                               (8+r)²=r²+12²

(a+b)²=a²+b²+2ab use the formula

                             64+r²+16r=r²+144

Cancel r² each other.

                             64+16r=144

                                    16r=144-64

                                     16r=80

                                         r=5

Radius of the circle    r=5cm

Therefore , the value of BC=2AC

                                       BC=2r

                                        BC=2*5

                                        BC=10cm

The length of line BC is 10cm

Part-2

To find out area of the circle

We have the formula for area of circle.

                                       Area=πr²

                                         Area=(22/7)*5*5

                                         Area=78.5 cm²

The area of the circle is 78.5 cm².

To find out the circumstance of Circle

We have the formula for circumstance of circle.  2πr

                              circumstance=2*(22/7)*5

                              circumstance=31.4 cm

The circumstance of Circle  is  31.4 cm.  

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In interval notation, the domain can be written as: (-∞, -5) ∪ (-5, 5) ∪ (5, ∞). In set notation, the domain can be written as: {x ∈ ℝ | x ≠ -5 and x ≠ 5}

To find the domain of the given rational expression, we need to determine the values of x that make the denominator equal to zero. This is because division by zero is undefined and therefore, these values of x cannot be included in the domain.

The denominator of the given expression is x^(2)-25. We can set this equal to zero and solve for x:

x^(2)-25=0

(x+5)(x-5)=0

x=-5 or x=5

These values of x make the denominator equal to zero, so they cannot be included in the domain. Therefore, the domain of the given rational expression is all real numbers except -5 and 5.

In interval notation, the domain can be written as: (-∞, -5) ∪ (-5, 5) ∪ (5, ∞)

In set notation, the domain can be written as: {x ∈ ℝ | x ≠ -5 and x ≠ 5}

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The two equations that might make up the system is  y = -7x + 16 and y = -2x - 9. Options C and E

Linear equations are defined as equations having its highest degree as 1.

From the information given, we have that;

The solutions are; (5, - 19)

From the equation(1):

y = -3x - 6

substitute the value of x

y = -3(5) - 6

y = -21

Equation (3)

y = -7(5) + 16

expand the bracket

y = -35 + 16

y = -19

y = -2x - 9

substitute the value

y= -2(5) - 9

y = -19

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The area of the yellow part of the pattern is found as: 72 cm²

The dimension of the square: 12 x 12

Area of yellow part = Area of triangle with base height equals side of square.

Then,

Area of yellow part = 1/2 * B * H

Area of yellow part = 1/2 *12 *12

Area of yellow part = 72 cm²

Thus, the area of the yellow part of the pattern is found as: 72 cm².

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The correct question is-

Work out the area of the yellow part of the pattern. This is a pattern that is being used to make a patch quilt. The pattern is a square, measuring 12cm

by 12cm square.

If x is the number of months the fee was charged, then the inequality to determine at most months the fee was charged is x ≥ (y + 46.83)/5.95  .

Let x be the number of months the fee was charged.

Let y be the initial balance before the fee was charged.

So, we can write the inequality as : y - 5.95x ≤ -46.83

where y is = positive number representing the initial balance.

Since the initial balance is "y" , we can determine the value of x that satisfies this inequality.

The inequality is represented as : x ≥ (y + 46.83)/5.95 .

Therefore, the inequality to determine number of months is x ≥ (y + 46.83)/5.95.

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The total cost of the services from Happy Paws and Woof Watchers will be equal if the dog is kept for 12 hours.

Finding the answer to a linear equation with one, two, three, or more variables is known as solving a linear equation. A linear equation's solution is, to put it simply, the value or values of the variables that make up the equation. Any coefficient can be deleted using the elimination procedure after being initially equated. Elimination is followed by equations being solved to get the other equation.

Let us suppose the number of hours = h.

Given that, Happy Paws charges $18.00 plus $2.50 per hour.

H = 18 + 2.50h

Woof Watchers charges $12.00 plus $3.75 per hour.

W = 12 + 3.75h

The cost is equal for:

18.00 + ($2.50 x h) = $12.00 + ($3.75 x h)

18 - 12 = 3.75h - 2.50h

h = 12

Hence, the total cost of the services from Happy Paws and Woof Watchers will be equal if the dog is kept for 12 hours.

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The coordinates to plot a graph are (2,0.19), (42,4), (63,6).

A graph is a visual representation of data that shows the relationship between variables. It consists of a set of points or vertices, connected by lines or curves called edges or arcs. Graphs are commonly used in mathematics, statistics, and other fields to display information in a way that is easy to understand and interpret.

There are several different types of graphs, including:

Line Graphs: Line graphs show the relationship between two variables using a continuous line. They are often used to display trends over time, such as stock prices or temperature changes.

Bar Graphs: Bar graphs use bars of different lengths to represent data. They are often used to compare different sets of data or to show the frequency of different values in a data set.

TOSS            DOLLARS

2                   0.19 (rounded to nearest cent)

42                 4

63                 6

To plot the points on a coordinate axes, you can use the TOSS values on the x-axis and the DOLLARS values on the y-axis. For example, the point (2, 0.19) would be plotted with 2 on the x-axis and 0.19 on the y-axis. Similarly, the point (42, 4) would be plotted with 42 on the x-axis and 4 on the y-axis, and the point (63, 6) would be plotted with 63 on the x-axis and 6 on the y-axis.

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The required gradient and  y-intercept for the given lines is given by ,

1. gradient = 10 and y-intercept = 20

2.  gradient = -20 and y-intercept = 10

3.  gradient = 0.5 and y-intercept = 2.5

Equation of a line in slope-intercept form = y = mx + c.

where 'm' is the slope of the line

And 'c' is the y-intercept.

For the line Y = 10x + 20

Compare with standard form we get,

Slope of the line is 10 .

And the y-intercept is 20.

This implies,

gradient is 10 and the y-intercept is (0, 20).

For the line  Y=10-20x

Compare with standard form we get,

Slope of the line is -20

And the y-intercept is 10.

This implies,

the gradient = -20

And the y-intercept = (0, 10).

For the line  Y=-2. 5+0. 5x

Compare with standard form we get,

Slope of the line is 0.5

And the y-intercept is -2.5

This implies,

the gradient = 0.5

And the y-intercept = (0, -2.5)

Therefore, the gradient and the y-intercept for each line is equal to,

Y=10x+20 , gradient = 10 and y-intercept = 20

Y=10 -20x  , gradient = -20 and y-intercept = 10

Y=-2. 5+0. 5x , gradient = 0.5 and y-intercept = 2.5

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Answer: x = p/2

Step-by-step explanation:

x = next one
p = previous

in order to solve the next one, you would do :
x (the one you are trying to solve) = p (the previous one : 1/8) divided by 2
x = p/2

The axis of symmetry for the function f(x) = -(x-7)² - 30 is x = 0.

A relation is a function if it has only One y-value for each x-value.

To find the axis of symmetry for the function f(x) = -(x-7)²- 30

We can use the formula:

x = -b/2a

where a and b are the coefficients of the quadratic term and the linear term in the function, respectively.

In this case, the quadratic term is -(x-7)² and the linear term is -30, so a = -1 and b = 0.

Substituting these values into the formula, we get:

x = -0/(2(-1)) = 0

Therefore, the axis of symmetry for the function f(x) = -(x-7)² - 30 is x = 0.

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The required matches for the given figures are 1.HI || EG, 2. AC⊥DB and ∠KFJ.

Orientation of the one line with respect to the horizontal or other respective line is known as a measure of orientation and this measure is known as the angle.

Here,
In the figure, we can see.

Line HI is parallel to line Eg so the correct choice is HI || EG.

Line AC is perpendicular to line AC, so the correct choice is AC⊥DB.
Angle made by points KFJ is ∠KFJ. so the correct choice is ∠KFJ.

Thus, the required matches for the given figures are 1.HI || EG, 2. AC⊥DB and ∠KFJ.

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It will take approximately 2.9 years (or 2 years and 11 months) for the $2500 investment to earn $400 interest at a rate of 8% compounded quarterly.

A = [tex]P(1 + r/n)^{nt}[/tex]

Where:

A = the total amount that includes principal and interest, P = the principal that is the initial amount invested, r = the annual interest rate, n = the number of times is the interest is compounded per year, and t = the time

$2900 = [tex]$2500(1 + 0.08/4)^{4t}[/tex]

Simplifying this equation, we get:

1.16 = (1 + 0.02

By calculating the natural logarithm of both sides, we arrive at:

ln(1.16) = 4t ln(1.02)

t ≈ 2.9 years

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The measure of angle x is 19.5

Octagon interior angles sum is equal to 1080 degrees. Also, the sum of all eight exterior angles is equal to 360 degrees.

We have Polygon of 8 side called Octagon.

Also, a Regular Octagon have all of its angles equal

So, Using Angle Sum Property

sum of all internal angle of Octagon  = 1080

8 x 6(x+3)= 1080

6(x+3) = 1080/8

6x+ 18 = 135

6x = 135- 18

6x = 117

x= 19.5

Thus, the value of x is 19.5

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Answer:

20.50 + 10m

Step-by-step explanation:

Basic expression:
8.50 + 12 + 6.25m + 3.75m

First, let's add the base payments which are 8.50 and 12.
8.50 + 12 = 20.50.
Now we have 20.50 + 6.25m + 3.75m

Next we have to add the monthly payments.
Assuming she's paying these all at the same time, 6.25m + 3.75m = 10m

Our finalized expression is:
20.50 + 10m = Club costs

The x number for which the following functions are equivalent is determined to be 3 by the preceding statement.

The term "function" refers to the correlation between a set of inputs and outputs. Simply defined, a function is an input-output relationship where each input is coupled to exactly one output.

The given functions are f(x) = x - 2 and f(x) = -2x + 7.

Assign the functions the following equivalences to determine the value of x:

variable x - 2 = -2x + 7

variable x + 2x = 7 + 2

Therefore, x = 3

Hence, the specified value for x is 3.

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The measure of angle ABC is 64°.

The angle bisector in geometry is the ray, line, or segment which divides a given angle into two equal parts.

Given that, BM bisects ∠ABC and ∠MBC=32°.

Here, ∠ABC=∠MBC+∠MBA

Since, BM bisects ∠ABC, ∠MBC=∠MBA

∠ABC=∠MBC+∠MBC

∠ABC=2∠MBC

∠ABC=2×32°

∠ABC=64°

Therefore, the measure of angle ABC is 64°.

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