The expanded quadratic equation with real coefficients is y = x² + 14 · x + 45.
How to determine the least polynomial that contains a given root
In this problem we need to determine the expanded quadratic equation with real coefficients such that one of its roots is - 7 + i 2. According with the quadratic formula, quadratic equations can have two conjugated complex roots, that is:
r₁ = α + β, r₂ = α - β
Then, the complete set of roots of the quadratic equation are r₁ = - 7 + i 2 and r₂ = - 7 - i 2. Then, the factor form of the polynomial is:
y = (x + 7 - i 2) · (x + 7 + i 2)
y = x · (x + 7 + i 2) + (7 - i 2) · (x + 7 + i 2)
y = x² + 7 · x + i 2 · x + (7 - i 2) · x + 7 · (7 - i 2) + i 2 · (7 - i 2)
y = x² + 7 · x + i 2 · x + 7 · x - i 2 · x + 49 - i 14 + i 14 - i² 4
y = x² + 14 · x + 45
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4. Describe the pattern. Choose the correct answer below. O A. The difference in the powers is equal to the sum of the bases. OB. The difference in the powers is equal to the difference in the bases. O C. The sum of the powers is equal to the sum of the bases. OD. There is no pattern.
Problem
Solution
Lets analyze one by one the possible options and we have:
O A. The difference in the powers is equal to the sum of the bases.
FALSE 2^2 and 1^2, we have 2-2=0 is different from 2+1=3
OB. The difference in the powers is equal to the difference in the bases.
FALSE , again if if we check 2^2 and 1^2 we have 2-1=1 and 2-2=0 and are not equal
O C. The sum of the powers is equal to the sum of the bases.
FALSE if we check 2^2 and 1^2 we have 2+1=3 and 2+2=4 and are not equal
OD. There is no pattern.
The best choice would be this one
Hello! I need help with this questionTriangle DOG was rotated to create triangle D'O'G''. Describe the transformation using details and degrees.
Answer:
180 degrees rotation about the origin
Explanation:
First, we need to identify the coordinates of DOG and D'O'G' as follows
D(-2, 1) ---> D'(2, -1)
O(-3, 3) ---> O'(3, -3)
G(1, 1) ---> G'(-1, -1)
Therefore, the rule for the transformation is
(x, y) ---> (-x, -y)
This rule is the rule for a 180 degrees rotation about the origin. So, the transformation is a 180 degrees rotation about the origin.
A teenager in a highrise building looks out his window at another building across the street and realizes a crime is being committed. If the buildings are 60 feet apart, and the teenager is looking down into the other building with an angle of depression of 20°, how many feet lower is the floor where the crime is being committed? If each floor is 10 feet, about how many floors down is it?
Hello!
The first step to solving this exercise is to draw the situation, look:
Now we can solve this exercise. Notice that we want to know the height that I called "X".
To find it, we must use the tangent of 20º, look:
[tex]\tan (20\degree)=\frac{opposite}{adjacent}[/tex][tex]\begin{gathered} \frac{0.364}{1}=\frac{60}{x} \\ \\ 0.364x=60 \\ x=\frac{60}{0.364} \\ \\ x\cong164.83 \end{gathered}[/tex]If each floor is 10 feet:
[tex]\frac{164.83}{10}=16.483\text{ floors}[/tex]If you have to approximate, you can say that it is 16.5 floors down.
Write an equation and graph the cosine function with amplitude 3 and period 4 pi.
An equation and graph the cosine function with amplitude 3 and period 4 pi is y is 3cos([tex]\frac{1}{2} x[/tex])
How to write an equation ?
The general form for the cosine function is:
y = A cos(Bx+C)+D
The amplitude is:
|A|
The period is:
P = 2π/B
The phase shift is
ϕ = −C/B
The vertical shift is D
Explanation:
Given:
The amplitude is 3:
|A|=3
The above implies that A could be either positive or negative but we always choose the positive value because the negative value introduces a phase shift:
A=3
Given:
The period is
P = 4π
4π =2π/B
B = 1/2
Nothing is said about the phase shift and the vertical shift, therefore, we shall assume that
C and D are 0.
Substitute these values into the general form:
y = 3cos([tex]\frac{1}{2} x[/tex])
An equation and graph the cosine function with amplitude 3 and period 4 pi is y is 3cos([tex]\frac{1}{2} x[/tex])
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The measure of angle 1 is greater than 97 degrees and at most 115. Graph the possible values of x.
We are given the following:
Angle 1 is greater than 97 degrees and at most 115 degrees
The angle alternate to angle 1 is "(9x +7)"; alternate angles are congruent
Using the information above, we will develop the inequality shown below:
[tex]\begin{gathered} 97^{\circ}10 \\ x>10 \\ \\ \therefore x>10 \end{gathered}[/tex]We will proceed to the second portion of the inequality. We have:
[tex]\begin{gathered} (9x+7)\le115^{\circ} \\ 9x+7\le115 \\ \text{Subtract ''7'' from both sides, we have:} \\ 9x\le115-7 \\ 9x\le108 \\ \text{Divide both sides by ''9'', we have:} \\ \frac{9}{9}x\le\frac{108}{9} \\ x\le12 \\ \\ \therefore x\le12 \end{gathered}[/tex]We will combine both inequalities into one. We have it thus:
[tex]\begin{gathered} x>10 \\ x\le12 \\ \text{Combining the inequalities, we have:} \\ 10We will proceed to plot this inequality on the number line as shown below:For this assignment, leave all answers as simplified fractions.A bag contains letter tiles that spell the name of the state MISSISSIPPI. Find the theoretical probabilityof drawing one tile at random for each of the following.1. P(M)2. P(S)3. P)4. P(not M)5. P(not S)6. P(not 1)
ANSWER
[tex]\begin{gathered} 1)\frac{1}{11} \\ 2)\frac{4}{11} \\ 4)\frac{10}{11} \\ 5)\frac{7}{11} \end{gathered}[/tex]EXPLANATION
From the given question;
MISSISSIPPI has 11 words
The number of M is 1
The number of S is 4
Hence;
1.
[tex]P(M)=\frac{1}{11}[/tex]2. P(S);
[tex]P(S)=\frac{4}{11}[/tex]4. P(not M) ;
[tex]\begin{gathered} 1-P(M) \\ =1-\frac{1}{11} \\ =\frac{10}{11} \end{gathered}[/tex]5. P(not S);
[tex]\begin{gathered} P(not\text{ S})=1-P(S) \\ =1-\frac{4}{11} \\ =\frac{7}{11} \end{gathered}[/tex]What is the effect on the graph of f(x) = 1 when it is transformed to g(x) = 1 + 17? O A. The graph of f(x) is shifted 17 units up. O B. The graph of f(x) is shifted 17 units to the left. O C. The graph of f(x) is shifted 17 units to the right O D. The graph of f(x) is shifted 17 units down.
What is the effect on the graph of f(x) = 1 when it is transformed to g(x) = 1 + 17? O A. The graph of f(x) is shifted 17 units up. O B. The graph of f(x) is shifted 17 units to the left. O C. The graph of f(x) is shifted 17 units to the right O D. The graph of f(x) is shifted 17 units down.
we have that
f(x)=1 ----> is a horizontal line (y intercept is (0,1)
f(x)=1+17=18 -----> is a horizontal line (y -intercept is (0,18)
therefore
there are a translation 17 units up
the answer is the option AWhat is the volume of the figure below?9 feet14 feet5 feet30ft60ft90ft120ft
The length of rectangular pyramid is l = 5 feet.
The width of rectangular pyramid is w = 4 feet.
The height of rectangular pyramid is h = 9 feet.
The formula for the volume of rectangular pyramid is,
[tex]V=\frac{l\cdot w\cdot h}{3}[/tex]Since there are two rectangular pyramid so net volume of figure is,
[tex]V^{\prime}=\frac{2}{3}\cdot l\cdot w\cdot h[/tex]Substitute the values in the formula to obtain the volume of figure.
[tex]\begin{gathered} V^{\prime}=\frac{2}{3}\cdot5\cdot4\cdot9 \\ =10\cdot4\cdot3 \\ =120 \end{gathered}[/tex]So volume of figure is 120 feet cube.
supposed your friend was born, your friends parents deposited $2000 in an account paying 3.5% interest compound monthly. what will the account balance be after 15 years?
If we have a deposit of $2000, invested at an an annual interest rate of 3.5% compounded monthly, we can calculate the final value of that deposit in 15 years as:
[tex]FV=PV(1+\frac{r}{m})^{m\cdot n}[/tex]where:
FV: final value of the deposit.
PV: initial deposit (PV = 2000)
r: annual interest rate (r = 0.035)
m: subperiod (as the compound is monthly, and there are 12 months in a year, we have m = 12).
n: period (n=15)
Then, the expression gives us a value of:
[tex]\begin{gathered} FV=PV(1+\frac{r}{m})^{m\cdot n} \\ FV=2000(1+\frac{0.035}{12})^{12\cdot15} \\ FV\approx2000(1.002917)^{180} \\ FV\approx2000\cdot1.689 \\ FV\approx3378 \end{gathered}[/tex]Answer: the account balance after 15 years will be $3378.
Driving down a mountain, Bob Dean finds that he descends 3300 feet in elevation by the time he is 4.5 miles (horizontally) away from the high point on the mountain road. Find the sic
descent. (1 mile 5280 feet).
The slope is
(Type an integer or decimal rounded to two decimal places as needed.)
Answer:
[tex]-0.14[/tex]
Step-by-step explanation:
Slope is defined as the change in [tex]y[/tex] divided by the change in [tex]x[/tex].
The change in [tex]y[/tex] is a decrease of 3,300 feet.
The change in [tex]x[/tex] is an increase of 4.5 miles. Converting to feet so that both measurements are in identical units:
[tex]\frac{4.5\text{ mi}}{1}\times\frac{5280\text{ ft}}{1\text{ mi}}=23760\text{ ft}[/tex]
Therefore, the slope is:
[tex]\frac{\Delta{y}}{\Delta{x}}=\frac{-3300}{23760}=-\frac{5}{36}[/tex]
As a decimal rounded to two places:
[tex]-\frac{5}{36}\approx{-0.14}[/tex]
find the mean of the data sets. round answers to the nearest integer. 4. { 24,21,23,22,26,23,24,21} 5. { 3,3,4,3,3,2}
Let's begin by identifying key information given to us:
[tex]24,21,23,22,26,23,24,21[/tex]The mean of the data set is calculated by using the formula:
[tex]\begin{gathered} m=\frac{SumOfTerms}{NumberOfTerms} \\ m=\frac{\Sigma x}{n}=\frac{24+21+23+22+26+23+24+21}{8} \\ m=\frac{184}{8}=23 \\ m=23 \end{gathered}[/tex]Therefore, the mean of the data set is 23
A candle is 4 inches tall and burns at the rate of 0.6 inch per hour. If the height of the candle after x hours is 1.5 inches, write an equation to represent the situation. Then use this equation to find the expected number of hours in which the candle melted to 1.5 inches.I think the way to start this is x = 1.5 inches but I'm stuck.
Given the following question:
Candle = 4 inches tall
Burns at a rate of 0.6 inches per hour
x = hours
[tex]4-0.6x=1.5[/tex]4 representing the height of the candle
0.6 representing the rate the candle burns
x = the amount of time in hours the candle burns
1.5 = the height of the candle after it burned for x hours
[tex]\begin{gathered} 4-0.6x=1.5 \\ 0.6x=4-1.5 \\ 0.6x=2.5 \\ x=2.5\div0.6 \\ x=4.166 \\ =4.166\text{ hours} \end{gathered}[/tex]What criteria can you use to show these two triangles are congruent?choice:SSAAASHLSAS
ANSWER
SAS
EXPLANATION
SAS criterion is an acronym for Side-Angle-Side criterion. Two triangles are congruent if the two sides and included angle of one triangle are equal to the corresponding sides and included angle of the second triangle, according to this criterion.
Write the equation of a function that has the given characteristics.The graph of y = |x|, shifted 8 units upwardоа y = |x-81y = |x+81y = x| + 8y = |x| - 8
The given function is
[tex]y=|x|[/tex]The transformation is shifted 8 units upwards.
Remember, to move the function upwards we have to sum outside the absolute value bars.
Therefore, the transformed function would be
[tex]y=|x|+8[/tex]I need help, I’m not sure which one is the answer
Given the triangles are similar, the pair of corresponding sides are proportional.
Then,
[tex]\frac{AB}{ED}=\frac{BC}{EF}[/tex]Knowing that:
AB = 30
ED = 5
BC = 42
EF = x
Substituting the values:
[tex]\begin{gathered} \frac{30}{5}=\frac{42}{x} \\ 6=\frac{42}{x} \end{gathered}[/tex]Multiplying both sides by x and then dividing the sides by 6.
[tex]\begin{gathered} 6*x=\frac{42}{x}*x \\ 6x=42 \\ \frac{6x}{6}=\frac{42}{6} \\ x=7 \end{gathered}[/tex]Answer: x = 7.
Jeff's restaurant sells hamburgers. The amount
charged for a hamburger (h) is based on the cost for a plain hamburger plus an additional charge for each topping (t) as shown in the equation below.
h = 0.60t+5
What does the number 0.60 represent in the equation?
A. the number of toppings
B. the cost of a plain hamburger
C. the additional cost for each topping
D. the cost of a hamburger with 1 topping
Answer:
Step-by-step explanation:
b the cost of the plain burger
not really sure what to do here??
Answer:
[tex]d=-6[/tex]
[tex]a_9=-49[/tex]
Step-by-step explanation:
If the first term of a sequence is [tex]-1[/tex] and the second term is [tex]-7[/tex], then each successive term in the sequence is [tex]6[/tex] less than the previous term. The common difference is [tex]-6[/tex] ( [tex]d=-6[/tex] ).
The formula representing the arithmetic sequence is:
[tex]a_n=-6n+5[/tex]
Therefore, the ninth term in the sequence is:
[tex]a_9=-6(9)+5[/tex]
[tex]a_9=-54+5[/tex]
[tex]a_9=-49[/tex]
can someone help me with this math problem a write a number line to it
Tom's Bakery recently spent a total of $800 on new equipment, and their average hourlyoperating costs are $10. Their average hourly receipts are $90. The bakery will soon makeback the amount it invested in equipment. What would the total expenses and receipts bothequal?TimeelapsedWrite a system of equations, graph them, and type the solution1.000SmartScore
Hello there. To solve this question, let's call the average hourly operating costs as C and the average hourly receipts as R.
We get that 90R - 10C = 800, since this is the amount the bakery has spent on equipments and it is said he will soon get back this invested money.
Dividing both sides by 10, we get 9R-C = 80
The graph will be a line. We want to find then C = R, so we can get:
-R + 9R = 80
8R = 80
R = 10
So, we have a cost of $100 and receipt of $900 in order to get back that money.
15/2 as a mixed number
We must express the number 15/2 as a mixed number, which is a number consisting of an integer and a proper fraction. To do that, we compute the quotient in the following way:
[tex]\frac{15}{2}=\frac{14+1}{2}=\frac{14}{2}+\frac{1}{2}=7+\frac{1}{2}=7\frac{1}{2}\text{.}[/tex]Answer
The number 15/2 expressed as a mixed number is:
[tex]7\frac{1}{2}\text{.}[/tex]Determine the amplitude, period, and phase shift for y=1/3tan (0 +30) and use them to plot the graph of the function.
Given: The function below
[tex]y=\frac{1}{3}tan(\theta+30^0)[/tex]To Determine: The amplitude, the period, and the phase shift
Solution
The graph of the function is as shown below
The general equation of a tangent function is
[tex]f(x)=Atan(Bx+C)+D[/tex]Where
[tex]\begin{gathered} A=Amplitude \\ Period=\frac{\pi}{B} \\ Phase-shift=-\frac{C}{B} \\ Vertical-shift=D \end{gathered}[/tex]Let us compare the general form to the given
[tex]\begin{gathered} y=\frac{1}{3}tan(\theta+30^0) \\ f(x)=Atan(B\theta+C)+D \\ A=\frac{1}{3} \\ B=1 \\ C=30^0 \\ D=0 \end{gathered}[/tex]Therefore
[tex]\begin{gathered} Amplitude=\frac{1}{3} \\ Period=\frac{\pi}{B}=\frac{180^0}{1}=180^0 \\ Phase-shift=-\frac{C}{B}=-\frac{30^0}{1}=-30^0 \end{gathered}[/tex]Hence, the correct option is as shown below
write an inequality: n is at most 40
If a value is at most 40, it means that such value cannot be greater than 40. The values will always take a value that is less tha 40 wih 40 inclusive.
To express this using inequality, we will use the less than or equal to sign in our expression as shown
[tex]n\leq40[/tex]You have one type of chocolate that sells for $5.00/lb and another type of chocolate that sells for $7.60/lb. You would like to have 10.4 lbs of a chocolate mixture that sells for $6.50/lb. How much of each chocolate will you need to obtain the desired mixture? You will need BLANK lbs of the cheaper chocolate and BLANK lbs of the expensive chocolate.Please help!! It is urgent.
Answer
You will need 4.4 lbs of the cheaper chocolate and 6 lbs of the expensive chocolate.
Explanation
Let the amount to be made of the type of chocolate (cheaper chocolate) that sells for $5.00/lb be x pounds.
Let the amount to be made of the type of chocolate (expensive chocolate) that sells for $7.60/lb be y pounds.
We would like to make a chocolate mixture that weighs 10.4 lbs. That is,
x + y = 10.4 .......... equation 1
This chocolate mixture is supposed to cost $6.50/lb and weigh 10.4 lbs. Hence, the cost of 10.4 lbs of this chocolate mixture will be
= 6.50 × 10.4 = $67.6
The cost of x pounds of the $5.00/lb = x × 5 = 5x dollars
The cost of y pounds of the $7.60/lb = y × 7.60 = 7.6y dollars
The cost of the two types of chocolate have to amount to $67.6
5x + 7.6y = 67.6 .......... equation 2
Writing the two equations together
x + y = 10.4
5x + 7.6y = 67.6
This simultaneous equation is then solved and the solution obtained is
x = 4.4 pounds and y = 6 pounds.
Hope this Helps!!!
Mira was solving for y in the following problem l. However, she did something wrong. Describe and correct the error in the work shown.2x-y=5y=-2x+5
The initial equation is:
[tex]2x-y=5[/tex]Then, in order to isolate y, we need to subtract 2x from each side of the equation:
[tex]\begin{gathered} 2x-y-2x=5-2x \\ -y=-2x+5 \end{gathered}[/tex]Then, we multiply both sides by -1:
[tex]y=2x-5[/tex]The equation Mira wrote is y=-2x+5. So we can see that her error was not changing the signs of the terms in the right side of the equation.
In order to correct her error, we just need to change the signs in the right side of the equation:
[tex]\begin{gathered} y=-(-2x+5) \\ y=2x-5 \end{gathered}[/tex]hi! i was wondering how to find the volume of a figure?
Answer: Volume of solid = 700 ft^3
Explanation:
The solid figure can be sectioned into 2 parts; a rectangular prism and a triangular prism.
Considering the rectangular prism, the formula for calculating the volume is
Volume = length x width x height
From the diagram,
length 10
height = 10
width = 5
Volume of rectangular prism = 10 x 10 x 5 = 500
Considering the triangular prism, the formula for calculating the volume is
Volume = base area x length
The base is a triangle
Area of triangle = 1/2 base x height
From the diagram,
base = 10
height = 8
Base area = 1/2 x 10 x 8 = 40
Length = 5
Volume of triangular prism = 40 x 5 = 200
Thus,
Volume of solid = 500 + 200
Volume of solid = 700 ft^3
The functions f and g are defined as follows:f(x)=2x^2-3xg(x)=-5x-1Find f(-3) and g(7)Simplify your answers as much as possible
f(x) = 2x^2 - 3x
f(-3) = 2(-3)^2 - 3(-3)
f(-3) = 2(9) + 9
f(-3) = 18 + 9
f(-3) = 27 This is the first solution
g(x) = -5x - 1
g(7) = -5(7) - 1
g(7) = -35 - 1
g(7) = -36 This is the second solution
FACTS:In 2010, the glacier was 1000 meters in length; it had retreated 255 meters in 15 years.Questions:1. Calculate the average annual rate of retreat of the glacier
Let's represent the given (length, years) as (x, y)
at time 0 year, the glacier is 1000 meters
at time 15 years, the glacier retreated 255 meters and that's 1000 - 255 = 745
so we have the points (0, 1000) and (15, 745)
Using the slope formula :
[tex]m=\frac{y_2-y_1_{}}{x_2-x_1}[/tex]where m represents the average annual rate of retreat.
[tex]m=\frac{745-1000}{15-0}=\frac{-255}{15}=-17[/tex]The average annual rate of retreat of the glacier is 17 meters per year
make your own quadratic function in standard form then determine the vertex of your propolis the axis of symmetry the y-intercept then graph using Circle curve line tools
The general form of a quadratic equation is expressed as
ax^2 + bx + c
An example of a quadratic equation is
2x^2 - 4x - 2
The graph of this equation is shown below
The vertex of the graph is value of x and y at the bottom of the curve. Looking at the curve, the vertex is (1, - 4)
The axis of symmetry is the vertical line that divides the curve into two equal halves. The axis of symmetry of this curve is the line, x = 1
The y intercept is the point where the curve cuts through the y axis. From the graph, y intercept = - 2
How many general tickets should the college sell to maximize revenue
Notice that, since 5000 tickets are reserved for students, there are 13000 available tickets that can be acquired by students or the general public.
Since the cost of a general public ticket is greater than that of a student ticket, we can increase the revenue to its maximum by selling the remaining 13000 tickets to the general public only.
Therefore, the answer is 13,000
Solve this system of equations byusing the elimination method.3x + 3y = 182x + y = 4( [?]. []).
Given the system of equations:
3x + 3y = 18
2x + y = 4
Let's solve the system of equations using the elimination method.
Multiply one equation by a number which makes one variable of each equation opposite.
Multiply equation 2 by -3:
3x + 3y = 18
-3(2x + y) = -3(4)
3x + 3y = 18
-6x - 3y = -12
Add both equations:
3x + 3y = 18
+ -6x - 3y = -12
_________________
-3x = 6
Divide both sides by -3:
[tex]\begin{gathered} \frac{-3x}{-3}=\frac{6}{-3} \\ \\ x=-2 \end{gathered}[/tex]Substitute -2 for x in either of the equations.
Take the second equation:
2x + y = 4
2(-2) + y = 4
-4 + y = 4
Add 4 to both sides:
-4 + 4 + y = 4 + 4
y = 8
Therefore, we have the solutions:
x = -2, y = 8
In point form, we have the solution:
(x, y) ==> (-2, 8)
ANSWER:
(-2, 8)