What is the least possible value of (x+1)(x+2)(x+3)(x+4)+2019 where x is a real number?
Answers
By completing the square, we have
[tex](x+1)(x+4) = x^2 + 5x + 4 = \left(x+\dfrac52\right)^2 - \dfrac94[/tex]
[tex](x+2)(x+3) = x^2 + 5x + 6 = \left(x + \dfrac52\right)^2 - \dfrac14[/tex]
Then
[tex](x+1)(x+2)(x+3)(x+4) = \left(x+\dfrac52\right)^4 - \dfrac52 \underbrace{\left(x + \dfrac52\right)^2}_{=y} + \dfrac9{16} \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = y^2 - \dfrac52 y + \dfrac9{16} \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \left(y - \dfrac54\right)^2 - 1 \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \left(\left(x + \dfrac52\right)^2 - \dfrac54\right)^2 - 1[/tex]
This product has a minimum of -1 when
[tex]\left(x + \dfrac52\right)^2 - \dfrac54 = 0[/tex]
which has two real solutions; then the minimum of the overall expression is -1 + 2019 = 2018.
Related Questions
Suppose that the functions and are defined as follows. p(x)=-x and q(x)=2x^2-2
find the following (q°p)(2)=
and (p°q)(2)=
Answers
The two compositions for the functions are:
[tex](p\°q)(2) = -6\\\\(q\°p)(2) = 6[/tex]
How to get the compositions?
When we have two functions q(x) and p(x), the composition is:
[tex](q\°p)(x) = q(p(x))[/tex]
Here we have:
[tex]p(x) = -x\\\\q(x) = 2x^2 - 2[/tex]
The first composition is:
[tex](q\°p)(2) = q(p(2)) = 2*(p(2))^2 - 2 = 2*(-2)^2 - 2 = 6[/tex]
The other composition is:
[tex](p\°q)(2) = p(q(2)) = -(q(2)) = -(2*2^2 - 2) = -(8 - 2) = -6[/tex]
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What is a possible value for the missing term of the geometric sequence? 25,_,874,…
Answers
Answer:
The middle term is 5√874======================
Given sequence25, x, 874, ...To findThe missing term xSolutionSince the given is the GP, find the common ratio in terms of given values:
r = x/25r = 874/xEquate the two values and solve for x to find the missing term:
x/ 25 = 874/xx² = 874*25x² = 21850x = √21850x = 5√874The sum of two numbers is 32. One number is 3 times as large as the other. What are the numbers?
Answers
Answer:
Let the greater number be x and the smaller number be y
A.T.Q,
x+y=32 ...(1)
x=3y ...(2)
putting the value of x from (2) in (1)
(3y)+y=32
4y=32
y= 32/4=8
x=3y
x=3(8)=24
x=24 , y=8
Dont Really need explanation Simply answer
Answers
Considering the given graph, the profit during the period was of -2 dollars.
How to add negative numbers along with positive?We add all the negative numbers, all the positive numbers, then keep the signal of the higher absolute value, and subtract the values.
From the graph, we have that:
The sum of the negative numbers is of 8.The sum of the positive numbers is of 2 + 4 = 6.8 > 6, hence the profit is negative given by:
-(8 - 6) = -2 dollars.
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point O, A, B lie on the same line. OA=12cm and OB=9cm. find the distance between the midpoints of segments OA and AB in each case
Answers
The Distance between the midpoints of segments OA and AB is 4.5cm
Lines and measuresA line is defined as the shortest distance between two points.
Given the following parameters
OA = 12cm
OB = 9cm
AB = OA - OB
AB = 12 - 9 = 3cm
midpoint of OA = 12/2 = 6cm
midpoint of AB = 3/2 = 1.5cm
Distance between the midpoints of segments OA and AB in each case will be the difference between the midpoints
Difference = 6cm - 1.5
Difference = 4.5cm
Hence the Distance between the midpoints of segments OA and AB is 4.5cm
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The height of a triangle is 4 m less than its base. The area of the triangle is 48 m². What is the length of the base? Enter your answer in the box.
Answers
The length of the base of the triangle is 12m
Area of a triangleThe formula for calculating the area of a triangle is expressed as:
A = 0.5 * base * height
If the height of a triangle is 4 m less than its base, then;
h = b - 4
Substitute
A = 0.5(b)(b-4)
48 = 0.5b²-2b
b²-4b - 96 = 0
Factorise
b² - 12b + 8b - 96 = 0
b(b-12)+8()b-12) = 0
b - 12 = 0
b = 12
Hence the length of the base of the triangle is 12m
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The boxed definition of absolute value states that |a|=-a if a is a negative number. Explain why |a| is always nonnegative, even though |a|=-a for negative values of a.
Answers
Final answer:
Since absolute values determine the distance between the number and the value whether the value is positive or negative. As distance is always positive.
Thus, |a| is always nonnegative, even though |a|=-a for negative values of a.
Step-by-step explanation:
Step 1
It is said that |a| is always nonnegative even though even though |a|=-a for negative values of a.
Step 2
This is because by the definition of an absolute value, any real number inside an absolute value symbol || will always be positive.
graph y(t) = 13.5t - 4.9t^2
Answers
Answer:
here
Step-by-step explanation:
Please help me....trying to get my HS diploma, i did not graduate :(
A ball is thrown in air and it's height, h(t) in feet, at any time, t in seconds, is represented by the equation h(t)=−t2+7t. When is the ball higher than 10 feet off the ground?
Answers
Answer:
A.
Step-by-step explanation:
so, the equation is
h(t) = -t² + 7t
so, we need to find the solutions for t (the time when the ball is exactly 10 ft in the air). there had to be 2 solutions, as the ball first goes up passing the 10 ft height, and then comes back down again, passing the 10 ft mark a second time. and between these 2 times the ball is higher (but not equal, so, we can only use < or > as inequality signs) than 10 ft.
10 = -t² + 7t
-t² + 7t - 10 = 0
the generation solution to a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = t
a = -1
b = 7
c = -10
t = (-7 ± sqrt(7² - 4×-1×-10))/(2×-1) =
= (-7 ± sqrt(49 - 40))/-2 = (-7 ± sqrt(9))/-2
t1 = (-7 + 3)/-2 = -4/-2 = 2 seconds
t2 = (-7 - 3)/-2 = -10/-2 = 5 seconds
so, between 2 and 5 seconds airtime the ball is higher than 10 ft.
and remember : HIGHER THAN.
so, we cannot use any equality (like <= or >=).
t must be higher than 2 and lower than 5 :
2 < t < 5
A baker purchased 13 lb of wheat flour and 14 lb of rye flour for a total cost of $14.85. A second purchase, at the same prices, included 15 lb of wheat flour and 12 lb of rye flour. The cost of the second purchase was $14.85. Find the cost per pound of the wheat flour and of the rye flour.
Answers
An equation is formed when two equal expressions. The cost per pound of wheat flour is $0.55 and the cost of per pound of rye flour is $0.55.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
Let the cost of a pound of wheat flour be x and the cost of a pound of rye flour be y.
A baker purchased 13 lb of wheat flour and 14 lb of rye flour for a total cost of $14.85.
13x + 14y = $14.85
x = (14.85-14y)/13
A second purchase, at the same prices, included 15 lb of wheat flour and 12 lb of rye flour.
15x + 12y = $14.85
Substitute the value of x from the above equation and solve for y,
[tex]\dfrac{15(14.85-14y)}{13} + 12y = 14.85\\\\222.75-210y+156y = 193.05\\\\y = 0.55[/tex]
Substitute the value of y in any one of the equations formed,
13x + 14(0.55) = 14.85
13x = 7.15
x = 0.55
Hence, the cost per pound of wheat flour is $0.55 and the cost of per pound of rye flour is $0.55.
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Transcribed image text: In 2000, 47% of the residents in a large city regularly used newspapers for getting news and this has decreased at an average rate of approximately 1.8% per year since then. Find a linear function in slope-intercept form that models this description. The function should model the percentage of residents, P(x), who regularly used the news outlet x years after 2000 P(x)= (Use integers or decimals for any numbers in the expression.)
Answers
Answer:
/.............................../././///////////////////////////////////////
Step-by-step explanation:
.
Help me please T^T
Solve for m.
m+1/3<-2 1/4
Answers
Answer:
answer is the second option m<-2 7/12
Step-by-step explanation:
process:
convert -2 1/4 into an improper fraction
m+1/3 < -9/4
m <-9/4 -1/3
calculate the difference
m<-31/12
turn into a mixed fraction
What is the sum of the first 20 terms of the arithmetic sequence 3,9,15,21,27
Answers
Answer: 6
Step-by-step explanation: 3 + 6 = 9, 9 + 6 = 15, 15 + 6 = 21, 21 + 6 = 27
Select the correct answer.
What is the solution to this equation?
(z
(x + 6) - 5 = -2
OA.
OB. 7
OC. 21
OD. -139
Answers
Answer:
-3
Step-by-step explanation:
Equation: (x + 6) - 5 = -2
we want to isolate x
(x + 6) - 5 = -2
+ 5 + 5 (add 5 to start getting x on its own)
x + 6 = 3
- 6 - 6 (subtract 6 to isolate x)
x = -3
so, this answer was not provided, but is the correct answer.
The solution to this equation is -3,
so, the correct answer is: -3.
Here, we have,
given that,
Equation is:
(x + 6) - 5 = -2
we want to isolate x
(add 5 to start getting x on its own)
(x + 6) - 5+ 5 = -2 + 5
(subtract 6 to isolate x)
x + 6 - 6 = 3 - 6
so, we get,
x = -3
so, the solution to this equation is -3,
so, the correct answer is: -3.
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Find the measure of the exterior 1.
A. 11°
OB. 127°
O C. 85°
O D. 95°
Answers
Step-by-Step Solution:
We know that the Exterior Angle of a Triangle = Sum of Opposite Interior Angles of the Triangle.
Therefore,
∠1 = 53° + 42°
=> ∠1 = 95°
Hence, ∠1 = 95°
Solve the following system by the substitution method. Check the solution(s)
7x + 4y = 10
x+y=1
Answers
Answer:
x = 2; y = -1
Step-by-step explanation:
We can solve (first for y, then for x) using substitution:
[tex]x+y=1\\7x+4y=10\\(x+y=1)-y\\x=1-y\\\\7(1-y)+4y=10\\7-7y+4y=10\\(7-3y=10)-7\\(-3y=3)/-3\\y=-1\\\\(x-1=1)+1\\x=2[/tex]
The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is four
times the measure of the first angle. The third angle is 12 more than the second. Let x, y, and z represent the measures of
the first, second, and third angles, respectively. Find the measures of the three angles.
Answers
Answer:
x = 36°
y = 66°
z = 78°
Step-by-step explanation:
General outline:Setup equations for each sentenceSolve the resulting systemSetup equations:
Allowing the first angle to be x, the second to be y, and the third to be z, the three sentences describing the relationships between angle measures are given below:
Eq1: x + y + z = 180°Eq2: y + z = 4xEq3: z = y + 12°Solve the systemSubstituting equation 3, into equations 1 and 2
Eq1: x + y + z = 180°
x + y + (y+12°) = 180°
x + 2y + 12° = 180°
x + 2y = 168°
x = 168° - 2y
Eq2: y + z = 4x
substituting the result from equation 1, and Equation 3...
y + (y+12°) = 4*(168° - 2y)
2y + 12° = 672° - 8y
10y = 660°
y = 66°
Substituting back into Equation 3...
Eq3: z = y + 12°
z = (66°) + 12°
z = 78°
Substituting both y, and z into Equation 1...
x + y + z = 180°
x + (66°) + (78°) = 180°
x + 144° = 180°
x = 36°
So, the three angle measures are:
x = 36°
y = 66°
z = 78°
Verification"The sum of the measures of the angles of a triangle is 180."
x + y + z = 180°
36° + 66° + 78° = 180°
True
"The sum of the measures of the second and third angles is four times the measure of the first angle"
y + z = 4x
66° + 78° = 4*(36°)
144° = 144°
True
"The third angle is 12 more than the second"
z = y + 12°
78° = 66° + 12°
True
Is the relationship between red sox away games and average daily revenues significant at the 95% confidence level? choose the correct answer with the corresponding correct reasoning.
Answers
Answer:
Yes, because the p-value of the independent variable is less than 0.05
Step-by-step explanation:
The relationship between Red Sox away games and average daily revenues significant at the 95% confidence level because the p-value ( X) < 0.05 , Option C is the correct answer.
What is Probability ?Probability is the study of chances of an event to happen.
It has a range form 0 to 1 , where 0 indicates uncertainty while 1 indicates certainty.
The data is given and on the basis of it
It can be seen that for 95 % confidence Level
P value =0.0005
So , the relationship between Red Sox away games and average daily revenues significant at the 95% confidence level because the p-value of the independent variable is less than 0.05.
Therefore Option C is the correct answer.
The complete question is
Is the relationship between Red Sox away games and average daily revenues significant at the 95% confidence level? Choose the correct answer with the corresponding correct reasoning.
No, because R Square is less than 50%
Yes, because the slope is positive.
Yes, because the p-value of the intercept is less than 0.05
Yes, because the p-value of the intercept is less than 0.95
Yes, because the p-value of the independent variable is less than 0.05
Yes, because the p-value of the independent variable is less than 0.95
The data table is attached with the answer.
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Question 9
The pOH of a solution is 5.70.
What is its hydroxide-ion concentration?
O 3.3 × 10-3
2.8 × 10-9
2.5 x 10-7
2.0 x 10-6
C
Answers
The pOH of a solution is 5.70. The correct option is C; 2.5 x 10-7 is the hydroxide ion concentration.
What is the pOH of the solution?The pOH of the solution is the negative logarithm of hydroxide ions concentration in a solution. The lower the value of pOH more alkaline will be the solution.
Mathematically written as:
pOH = - Log[OH-]
The sum of ph and pOH is equal to 14.
pH + pOH = 14
The pOH of a solution is 5.70.
pOH = - Log[OH-]
5.70 = - Log[OH-]
[OH-] = 2.5 x 10-7 is the hydroxide ion concentration.
The solution has a pH value of more than 7 which means that it is base.
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(1.5 x 10^8) + ( 2.06 x 10^8) =
Answers
(1.4x10^8) + (2.06x10^8)=
356,000,000
Answer:
356000000
Step-by-step explanation:
1. Simplify the expression
Calculate the expression inside the parentheses.
[tex](1.5*10^8)+(2.06*10^8)[/tex]
Simplify exponents and square roots
[tex](1.5*10^8)+(2.06*10^8)[/tex]
[tex](1.5*100000000)+(2.06*10^8)[/tex]
Perform any multiplication or division, from left to right:
[tex](1.5*100000000)+(2.06*10^8)\\[/tex]
[tex](150000000)+(2.06*10^8)[/tex]
Simplify exponents and square roots
[tex]150000000+(2.06*10^8)[/tex]
[tex]150000000+(2.06*100000000)[/tex]
Perform any multiplication or division, from left to right:
[tex]150000000+(2.06*100000000)[/tex]
[tex]150000000+(206000000)[/tex]
Calculate any addition or subtraction, from left to right.
[tex]150000000+206000000[/tex]
[tex]356000000[/tex]
Why learn this
Order of operations is a basic rule of algebra. It tells us what to solve first when we have an equation with multiple functions, which is something that you will likely encounter throughout your maths studies. The order is: Parentheses, Exponents, Multiplication and Division, then finally Addition and Subtraction. Remember that when you're solving within the parentheses, the Order of Operations applies!
Terms and topics
Order of operationsKate is the captain of a large cruise ship. As she
drives the ship, she sits 50 meters above the sea.
Paul is the captain of a boat. He sits 15 meters
above the sea. If the radius of the earth is
6,371,000 meters, how much further can Kate see
across the horizon than Paul?
here are the first few questions the rest are in the picture.
1.Show me that you understand the problem. restate it in your own words and identify the important information.
2. what is your prior knowledge and what tools do you have to solve this problem?
a. Draw two diagrams, one for Kate and one for Paul. Include important numbers. Insert your diagrams into your answer.
Answers
The amount further by which Kate can see across the horizon than Paul is; 3.33 times farther
How to use volume of a cone in applications?Formula for volume of a cone is;
V = ¹/₃πr²h
For Kate:
Height (h) = 50 m
Radius (r) = 6,371,000 m
Thus;
V = ¹/₃ * π * (6,371,000)² * 50
V = 2.12526863 × 10¹⁵ m³
To determine how much further can kate see across the horizon than Paul, we have:
⇒ (2.12526863 × 10¹⁵ m³)/6.3758059 × 10¹⁴ m³
= 3.33 times further.
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Natasha had a $922.93 balance on her credit card at the beginning of September. Her credit card has an APR of 9.89%, compounded monthly, and a minimum monthly payment of 3.08% of the total balance. The following table shows Natasha’s credit card purchases over the next two months.
Month
Cost ($)
September
33.70
October
61.70
October
27.80
If Natasha makes only the minimum payments, what will her balance at the beginning of November? (Assume that the interest accrues before the monthly payment, and that the monthly payment occurs at the end of the month. Round all dollar values to the nearest cent.)
Answers
Answer:
Her balance at the beginning of November=$973.76
Step-by-step explanation:
What is Natasha's credit balance at the beginning of November?(i.e. at the end of October)
The balance at the end of each month is the beginning balance plus the current month's purchases minus the minimum payment for the month.
Sept =$922.93+$33.70+9.89%/12*($922.93+$33.70)
Sept=$964.51
payment= 964.51*3.08%
balance at the end of Sept=$964.51-($964.51*3.08%)
balance at the end of Sept=$934.80
Oct=$934.80+$61.70+9.89%/12*($934.80+$61.70)
Oct=$ 1,004.71
payment=$1,004.71*3.08%
balance at the end of Oct(at beginning of Nov)=$ 1,004.71-($1,004.71*3.08%)
balance at the end of Oct(at beginning of Nov)=$973.76
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Answer:
The answer is C.$1,000.93 just did it om edge
Step-by-step explanation:
(02.02 LC)
If h(x) = −2x − 10, find h(−4).
Group of answer choices
−2
−18
−3
−16
Answers
The value of the function h(-4) is (a) -2
How to determine the h(-4)?The function is given as
h(x) = -2x - 10
When x = -4, we have:
h(-4) = -2 * -4 - 10
Evaluate the product
h(-4) = 8 - 10
Evaluate the difference
h(-4) = -2
Hence, the value of the function h(-4) is (a) -2
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A box contains 162 marbles which are either red or blue. There are 42 more blue marbles than red marbles. How many blue marbles are in the box? Blue marbles=
Answers
The number of the blue marbles are in the box will be 102.
What is the solution of the equation?The solution of the equation means the value of the unknown or variable.
A box contains 162 marbles which are either red (r) or blue (b).
r + b = 162 ...1
There are 42 more blue marbles than red marbles.
b = r + 42 ...2
Then from equation 1 and 2, we have
r + r + 42 = 162
2r = 120
r = 60
Then the number of blue marble will be
b = 60 + 42
b = 102
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Scientists want to test a new pair of running shoes. A speed test is performed with two separate groups of participants. The treatment group will wear the new pair of running shoes, while the control group will not. It is believed that age and height may affect speed. Which of the following would be most effective in controlling the confounding variables, such as age and height, in this study?
Answers
The most effective in controlling the confounding variables is matched pairs design.
What is a matched pair design experiment?A matched pairs design is when participants are matched based on shared characteristics that is of importance to the experiment. One member of the pair is put in the control group while the other is put in the experimental group.
Here are the options:
A completely randomized design experiment
A longitudinal observational study
A retrospective observational study
A matched-pair design experiment
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solving 3 x plus 8 equals 14
Answers
3x + 8 = 14
To solve subtract 8 from both sides to isolate x
3x = 6
Now divide both sides by 3 to solve for x
X = 2
x, 3x and 5x ,find the sum
Answers
x + 3x + 5x ( the variables are same you can add them immediately )
9x ( the answer is 9x for adding terms with the same variable just add the numbers and write the variable as itself)
HOPE IT HELPS
Answer: 9x
Step-by-step explanation:
As the variables are of same power (1) we can add it
x+3x+5x=9x
Solve linear equations using substitution y=-4x
y=4x+16
Answers
Answer:
(-2 , 8)
Step-by-step explanation:
[tex]\begin{cases}y=-4x&\\ y=4x+16&\end{cases}[/tex]
We substitute y by -4x in the second equation of the system:
-4x = 4x + 16
⇔ -8x = 16
⇔ x = 16/(-8)
⇔ x = -2
Now ,we substitute x by -2 in the first equation of the system:
y = -4 × (-2) = 8
then
the solution of the system is {(x = -2 , y = 8)}
How many significant figures does 119.3 have
Answers
Not example 155.000
Choose the number sentence that applies the commutative property to the example.
3 × 4 = 12
6 + 6 = 12
12 × 1 = 12
6 × 2 = 12
4 × 3 = 12
Answers
commutative property in multiplication is
a•b = b•a