The given quadratic function, y=x^2+7x+4, in vertex form is y = (x + 7/2)^2 - 33/4
Writing Quadratic functions in Vertex formFrom the question, we are to write the given quadratic function in vertex form
To write the quadratic function y = x^2 + 7x + 4 in vertex form, we can complete the square. The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex.
To complete the square, we need to add and subtract a constant inside the parentheses so that the expression inside the parentheses becomes a perfect square trinomial. We can do this by adding and subtracting (7/2)^2 = 49/4 inside the parentheses:
y = x^2 + 7x + 4
= (x^2 + 7x + 49/4) - 49/4 + 4 (adding and subtracting 49/4)
= (x + 7/2)^2 - 33/4
Hence, the vertex form is y = (x + 7/2)^2 - 33/4
NOTE: The vertex is (-7/2, -33/4).
Here is the correct question:
Write the quadrat function Y=x^2+7x+4 in vertex form
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Which point represents (3, negative StartFraction 5 pi Over 3 EndFraction) ?
WILL GIVE BRAINLIEST
The point (3, -5π/3) can be represented by the formula x = 3, y = -5π/3.
The point (3, -5π/3) can be represented by the formula x = 3, y = -5π/3. To calculate this, we first find the x-coordinate, which is simply 3. To find the y-coordinate, we must first multiply 5 and π. 5π = 15.78. We then divide 15.78 by 3, which gives us 5.26. To make this a negative number, we add a negative sign, giving us -5.26. We can then convert this to fraction form by dividing the numerator and denominator by the greatest common factor, which is 1. This gives us -5π/3.
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Complete question:
What point represents (3, negative StartFraction 5 pi Over 3 EndFraction)?
Answer:
A
Step-by-step explanation:
Triangle
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△A
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B
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C
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triangle, A, prime, B, prime, C, prime is the result of dilating
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△ABCtriangle, A, B, C about point
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PP by a scale factor of
3
33.
The first quadrant of a coordinate plane. The x- and y-axes both scale by one. A pre-image Triangle A B C has point A at ten, six, point B at thirteen, seven, and Point C at seven, twelve. A point P is at nine, eight.
The first quadrant of a coordinate plane. The x- and y-axes both scale by one. A pre-image Triangle A B C has point A at ten, six, point B at thirteen, seven, and Point C at seven, twelve. A point P is at nine, eight.
Determine whether each claim about the properties of
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△ABCtriangle, A, B, C and
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triangle, A, prime, B, prime, C, prime is true or false.
Line segments AB and A'B' are on the same line: False.
Line segments AC and A'C' are on parallel: False.
What is dilation?In Geometry, dilation can be defined as a type of transformation which typically changes the size of a geometric object, but not its shape.
Based on the transformation rule, line segment AB would move further away from point P because both coordinates A and B move away from point P. Therefore, line segments AB and A'B' cannot be on the same line because they are parallel.
Conversely, line segment AC and A'C' are located on the same line because points A and C would move away from P, but line segment AC passes through point P, and line segment A'C' would do the same. Therefore, line segments AC and A'C' are not parallel.
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Answer:
true false
Step-by-step explanation:
x²+6x-3=0
but how do you find x=0 for two brackets?
eg- (x+?)=0 x=? and (x+?)=0 x=?
The factored form of the quadratic function x² + 6x - 3 = 0 is given as follows:
x² + 6x - 3 = (x + 6.46)(x - 0.46).
How to write the factored form of the quadratic function?The quadratic function for this problem is defined as follows:
x² + 6x - 3 = 0.
The coefficients are given as follows:
a = 1, b = 6, c = -3.
Hence the discriminant of the quadratic function is obtained as follows:
D = b² - 4ac
D = 6² - 4(1)(-3)
D = 48.
Then the first root is given as follows:
x = (-b - sqrt(D))/2a
x = (-6 - sqrt(48))/2
x = -6.46.
The second root is then obtained as follows:
x = (-b + sqrt(D))/2a
x = (-6 + sqrt(48))/2
x = 0.46.
Meaning that the factored form of the function is given as follows:
x² + 6x - 3 = (x + 6.46)(x - 0.46).
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The height of a cylinder is 1 inch less than the diameter of the cylinder. Which expression represents the volume of the cylinder in terms of its radius, x?
Answer:
A
Step-by-step explanation:
given the radius is x , then diameter = 2x and height = 2x - 1
the volume (V) of a cylinder is calculated as
V = πr²h ( r is the radius and h the height ) , then
V = πx²(2x - 1) ← distribute parenthesis by πx²
= 2πx³ - πx² in³
What is the slope of the line that passes through the points (-4, 6)and(-4,18)
Answer:
Undefined
Step-by-step explanation:
The Xs are the same!
100 points please help
The factors of the expression are (2x + 3) (2x - 9). The solution has been obtained by using factorization.
What is factorization?
Writing a number or other mathematical object as the result of numerous factors, typically smaller or simpler terms of the same kind, is known as factorization or factoring in mathematics.
We are given an expression as 4[tex]x^{2}[/tex] - 12x -27.
On factoring it, we get
⇒ 4[tex]x^{2}[/tex] - 12x -27
⇒ 4[tex]x^{2}[/tex] + 6x - 18x -27
⇒ 2x (2x + 3) - 9 (2x + 3)
⇒ (2x + 3) (2x - 9)
Hence, the factors of the expression are (2x + 3) (2x - 9).
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Company ABC is offering bonds to investors to pay for its corporate
expansion.
• Par value: $1,000 per bond
Coupon rate: 5 percent per year (fixed rate)
• Maturity: 10 years
Each year the bond would pay the bondholder the following amount:
$500.
$50.
$5.
$1,500.
$1,000.
The bond would pay the bondholder the following sum of $1,000 annually.
The bond being offered by Company ABC has a par value of $1,000, a fixed coupon rate of 5% per year, and a maturity of 10 years.
The coupon rate of 5% per year means that for each $1,000 bond, the bondholder will receive an annual interest payment of $50 (5% of $1,000).
Therefore, out of the given options, the correct amount that the bondholder would receive each year is $50.
It's important to note that this interest payment is fixed and does not change over the 10-year term of the bond. At maturity, the bondholder would receive the par value of $1,000 back from the company.
So, over the course of 10 years, the bondholder would receive a total of $500 ($50 per year) in interest payments, in addition to the return of the $1,000 par value at maturity.
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select the two values of x that are roots of this equation 2x²-11x+15=0
Answer:
[tex]x=\frac{5}{2},\:x=3[/tex]
Step-by-step explanation:
[tex]\mathrm{Factor\:}2x^2-11x+15[/tex]
[tex]\mathrm{Break\:the\:expression\:into\:groups}[/tex]
[tex]\left(2x^2-5x\right)+\left(-6x+15\right)[/tex]
[tex]\mathrm{Factor\:out}\:x\:\mathrm{from}\:2x^2-5x:\quad \:x\left(2x-5\right)[/tex][tex]\mathrm{Factor\:out}\:-3\:\mathrm{from}\:-6x+15:\quad \:-3\left(2x-5\right)[/tex]
[tex]x\left(2x-5\right)-3\left(2x-5\right)[/tex]
[tex]\mathrm{Factor\:out\:common\:term\:}2x-5[/tex]
[tex]\left(2x-5\right)\left(x-3\right)[/tex]
[tex]\mathrm{Using\:the\:Zero\:Factor\:Principle:\quad \:If}\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0[/tex]
[tex]2x-5=0\quad \mathrm{or}\quad \:x-3=0[/tex]
[tex]\mathrm{Solve\:}\:2x-5=0:\quad x=\frac{5}{2}[/tex]
[tex]\mathrm{Solve\:}\:x-3=0:\quad x=3[/tex]
[tex]\mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:}[/tex]
[tex]x=\frac{5}{2},\:x=3[/tex]
Use Pythagorean theorem to find perimeter What is the perimeter of the triangle? in units PLS HELP
Answer:
30. Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
In 2022, a customer buys 5 GE 10% debentures, M '41. The interest payment dates are Feb 1st and Aug 1st. The current yield on the bonds is 11. 76%. The bonds are callable as of 2032 at 103. The bond is trading:
The current yield of the bond is 11.76% and the yield to call is 19.51%.
The current yield of the bond can be calculated using the following formula:
Current Yield = (Annual Interest Payment/Bond Price) * 100
In this case, the annual interest payment is 10% of the face value of the bond (5 * $100 = $500). The bond price is $41, so the current yield is 11.76%:
Current Yield = ($500/$41)*100 = 11.76%.
The bond is callable at 103 as of 2032, so the call premium is 103 – 41 = 62. This means that if the bond is called, the investor will receive an additional payment of $62 per bond (5 * $62 = $310). The yield to call is the total annual return the investor will receive if the bond is called:
Yield to Call = (Annual Interest Payment + Call Premium)/Bond Price * 100
In this case, the yield to call is:
Yield to Call = ($500 + $310)/$41 * 100 = 19.51%.
This means that if the bond is called in 10 years, the investor will receive an annual return of 19.51%.
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A music company is introducing a new line of acoustic guitars next quarter. these are the cost and revenue functions, where x represents the number of guitars to be manufactured and sold: r(x) = 120x, c(x) = 100x 1,840. the company needs to sell at least guitars for a total revenue of $______to start making a profit.
To reach the revenue needed to cover expenses and begin turning a profit, or 120x = 120(92) = $11,040, the business must sell at least 92 guitars.
To start making a profit, the music company needs to generate enough revenue to cover its costs. The revenue function is given by r(x) = 120x, where x is the number of guitars manufactured and sold.
The cost function is given by c(x) = 100x + 1840, where 1840 represents the fixed cost (such as rent, salaries, etc.) associated with setting up the manufacturing process.
To determine the total revenue required to start making a profit, we need to find the break-even point where the revenue generated is equal to the total cost incurred:
r(x) = c(x)
Substituting the given functions, we get:
120x = 100x + 1840
Solving for x, we get:
20x = 1840
x = 92
Therefore, the company needs to sell at least 92 guitars to generate a revenue of 120x = 120(92) = $11,040, which is the amount required to cover its costs and start making a profit. If the company sells fewer than 92 guitars, it will not be able to cover its costs and will incur a loss.
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Factor completely
72w^2-50w^4
hm, dunno.
2w^2(5w + 6)(-5w + 6)
Answer:
2w^2(6 + 5w)(6 - 5w)
Step-by-step explanation:
Write the LCM of the following A=a^3× b^2×c and B=a^5 ×b×c^4×d , giveyour answer in exponential form
The LCM of A = a^3× b^2×c and B=a^5 ×b×c^4×d in exponential form is a^5 × b^2 × c^4 × d^1.
To find the LCM (Least Common Multiple) of two numbers, we need to find the greatest power of each prime factor that occurs in either number.
The prime factors in the given expressions are a, b, c, and d. For each prime factor, we need to find the greatest power that occurs in either A or B.
For a: The highest power of a that occurs in A is a^3, and the highest power that occurs in B is a^5. Therefore, the LCM must include a^5.
For b: The highest power of b that occurs in A is b^2, and the highest power that occurs in B is b^1. Therefore, the LCM must include b^2.
For c: The highest power of c that occurs in A is c^1, and the highest power that occurs in B is c^4. Therefore, the LCM must include c^4.
For d: The highest power of d that occurs in A is d^0, and the highest power that occurs in B is d^1. Therefore, the LCM must include d^1.
Putting all of this together, we have:
LCM(A,B) = a^5 × b^2 × c^4 × d^1
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Calculate the pressure if the force is 270N/m2 and the area is 0.03m2
If the force is 270 Newton and the area is 0.03 square meter, the pressure is equal to 9,000 N/m².
What is pressure?In Mathematics, pressure can be defined as a measure of the force exerted per unit area of an object or physical body. This ultimately implies that, it is usually measured in Newton per meter square.
How to calculate the pressure of an object?Mathematically, the pressure of a physical object can be calculated by using this formula:
P = F/A
Where:
P represents the pressure.F represents the force.A represents the area.By substituting the given parameters, we have:
Pressure, P = 270/0.03
Pressure, P = 9,000 N/m².
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HELP PLEASE IM SO CONFUSED
Theater Club Stage Crew
The school's theater club is building sets that will make ordinary students look like giants. The actors need a door, a table, and a stool that will make them look almost twice as tall!
DOOR: actual height is 80 inches and Height on Set is 44 inches
TABLE: actual height is 28 inches
STOOL: actual height is 18 inches
The heights of the objects in the set are proportional to the actual heights of objects. What is the CONSTANT OF PROPORTIONALITY? Write as a whole number, decimal or fraction
The constant of proportionality of the relation between the height on set and the actual height is given as follows:
0.55.
What is a proportional relationship?A proportional relationship is defined according to the equation presented as follows:
y = kx.
In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.
The variables for this problem are given as follows:
y is the height on set.x is the actual height.Considering the values of these variables for the door, the constant is obtained as follows:
k = 44/80
k = 0.55.
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Find the length of side x in simplest radical form with a rational denominator.
Step-by-step explanation:
There are a couple of ways to solve this...this right triangle has two legs of length x and hypotenuse of 9 units
Use Pythagorean theorem:
x^2 + x^2 = 9^2
x = 9/sqrt2 = 9 sqrt (2) / 2
Milan is driving to Memphis. Suppose that the distance to his destination (in miles) is a linear function of his total driving time (in minutes). Milan has 45 miles to his destination after 39 minutes of driving, and he has 26.3 miles to his destination after 61 minutes of driving. How many miles will he have to his destination after 79 minutes of driving?
The distance after 79 minutes is 11 miles.
How many miles will he have to his destination after 79 minutes of driving?We know that the distance can be modeled with a linear equation like:
y = ax + b
Where a is the slope and b is the y-intercept.
We know that if the line passes through two points (x₁, y₁) and (x₂, y₂) then the slope can be written as:
a = (y₂ - y₁)/(x₂ - x₁)
Here we know the points (39, 45) and (61, 26.3)
Then we can write:
a = (26.3 - 45)/(61 - 39) = -0.85
So we can write:
y = -0.85*x + b
To find the value of b we can replace the values of one of the points, i will use (39, 45) to get.
45 = -0.85*39 + b
45 + 0.85*39 = b
78.15 = b
So the line is:
y = -0.85*x + 78.15
The distance after 79 minutes is.
y = -0.85*79 + 78.15
y = 11
The distance left is 11 miles.
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I neeeddd this answerrr asppp plsssss
Answer: the very top answer -> A
Two schools are investigating whether there is a difference in the proportion of students who attend the homecoming football game. Both schools have over 2,000 students. School A selected a simple random sample of 100 students and found that 98 attended the homecoming football game. School B selected a simple random sample of 150 students and found that 142 attended the homecoming football game. Have the conditions for statistical inference for testing a difference in population proportions been met
All four conditions have been met, the conditions for statistical inference for testing a difference in population proportions have been satisfied. The conditions for statistical inference for testing a difference in population proportions are:
Random samples: Both schools have selected simple random samples of their respective student populations, which satisfies this condition.
Large sample size: Both schools have over 2,000 students, which satisfies the large sample size condition.
Independent samples: The samples from schools A and B are independent, as the students from one school do not affect the students from the other school.
Success-failure condition: The sample sizes are large enough to satisfy the success-failure condition, which states that there should be at least 10 successes and 10 failures in each sample. Both schools have well over 10 successes and failures in their respective samples.
Since all four conditions have been met, the conditions for statistical inference for testing a difference in population proportions have been satisfied.
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In OH, IR = jk, mik = (11x + 2)", and mjk = (12x - 7).
What is the measure of ik)?
mik) =
Intro
Done
The measure of ik in the given equation can be calculated by subtracting the sum of IR (which is equal to jkh) and mjk from 180e degrees.
In order to calculate the measure of ik in the given equation, we must use the formula for interior angles of a triangle which is the sum of all angles in a triangle is 180 degrees. In this equation, we are given the measures of two of the angles, IR and mjk. Therefore, the measure of ik can be calculated by subtracting the sum of IR and mjk from 180 degrees.
Given that IR = jk, we can calculate the measure of ik by first calculating the measure of jk. We are given that mik = (11x + 2), so we can solve for x and use that value to calculate jk. If we solve for x, we find that x = (mik-2)/11. We can then use that value to calculate jk, which equals mik-mjk. In this equation, we are given that mjk = (12x - 7), so jk = (11x + 2) - (12x - 7). Simplifying, we find that jk = 9x + 9.
Now that we have the measure of jk, we can calculate the measure of ik by subtracting the sum of jk and IR from 180 degrees. Since IR = jk, we can substitute jk in for IR, so ik = 180
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1. Calculate the mean for each sample.
Sample #
1
234
2
3
5
6
7
8
9
O ACCESS Virtual Learning 2022
Sample Items
(1, 1, 2, 0)
(2, 2, 0, 1)
(1, 1, 1, 2)
(2,0, 1, 0)
(2, 2, 0, 1)
(1, 1, 2, 3)
(3, 2, 0, 0)
(1, 1, 1, 1)
(3, 1, 1, 1)
x
Step-by-step explanation:
1. Calculate the mean for each sample.
Sample #
1
234
2
3
5
6
7
8
9
O ACCESS Virtual Learning 2022
Sample Items
(1, 1, 2, 0)
(2, 2, 0, 1)
(1, 1, 1, 2)
(2,0, 1, 0)
(2, 2, 0, 1)
(1, 1, 2, 3)
(3, 2, 0, 0)
(1, 1, 1, 1)
(3, 1, 1, 1)
x
4. What value of b makes this proportion
true?
12 = 21
A. B= 25
C. B = 28
B. B= 27
D. B = 30
The answer is not provided among the options given.
An equation known as proportion shows that the two ratios given are equal to one another. In other terms, the proportion declares that the two ratios or fractions are equal.
To solve for the value of b that makes the proportion true, we can use cross-multiplication:
12 = 21A
12b = 21A
b = 21A/12
To determine the specific value of b, we need to know the value of A. However, since A is not given in the question, we cannot determine the value of b. Therefore, the answer is not provided among the options given.
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PRACTICE PROF
Problem A Jimmy is having a birthday party at the zoo. The zoo has a fixed fee
birthday parties, plusafee per person. Jimmy is told the total charge for 10 peo
including himself, would be $97.50 and the total charge for 20 people, includin
himself, would be $175. Determine the:
I
a. independent and dependent variables
b. rate of change
c. initial value d. the total charge for 17 people
e. the number of people who
a. The independent and dependent variables are:
Independent: The number of personDependent: The total chargeb. Rate of change = $7.75
c. initial value = $20
d. The total charge for 17 people = $151.75
e. The number of people who could come for $500 is $62.
How do we solve these problems?To determine the independent and dependent variables, we need to identify which variable depends on the other. In this case, the total charge depends on the number of people attending the party. Therefore, the number of people is the independent variable, and the total charge is the dependent variable.
To find the rate of change, we can use the slope formula:
slope = (change in y) / (change in x)
slope = (175 - 97.5) / (20 - 10)
slope = 7.75
Therefore, the rate of change is $7.75 per person.
To find the initial value, we can use either of the two data points. Let's use the data point where Jimmy and 10 people have a total charge of $97.50.
y = mx + b
97.5 = 7.75(10) + b
b = 20
Therefore, the initial value (the fixed fee for the party) is $20.
To find the number of people who could come for $500, we can rearrange the equation:
y = mx + b
500 = 7.75x + 20
480 = 7.75x
x ≈ 62
Therefore, Jimmy could invite 62 people (including himself) to the party for $500.
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Consider the expression . Which of the following is an equivalent expression ?
Answer:
4 th option bcoz [tex]x^{y+2}=x^{y}x^{2[/tex]
Points P,Q,S appear in that order on a line. The ratio PQ:QR is 3:4. The ratio QR:RS is 2:5. The length PQ is 6 in. Find the length
PS
The length of PS is 34 inches. Points P,Q,S are points on a line, so that PQ is 6 in, The ratio PQ:QR is 3:4 and The ratio QR:RS is 2:5.
How to solve an equation?An equation is an expression containing numbers and variables linked together by mathematical operations such as addition, subtraction, division, multiplication and exponents.
The length of PQ is 6 in. The ratio PQ:QR is 3:4, hence:
PQ = PR * (3/7)
6 = PR * (3/7)
PR = 14 inches
QR = PR * (4/7)
QR = 14 * 4/7
QR = 8 inches
The ratio QR:RS is 2:5, hence:
QR = QS * (2/7)
8 = QS * 2/7
QS = 28 inches
Points P,Q,S appear in that order on a line. Therefore:
PS = PQ + QS
PS = 6 + 28
PS = 34 inches
The length of PS is 34 inches.
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a survey of 400 college seniors resulted in the following crosstabulation regarding their undergraduate major and whether or not they plan to go to graduate school. undergraduate major graduate school business engineering other total yes 35 44 64 143 no 92 101 64 257 total 127 145 128 400 what percentage of the students' undergraduate major is engineering?
The percentage of college seniors' undergraduate major that is engineering is 36.25%.
To calculate the percentage of college seniors' undergraduate major that is engineering, we need to find the proportion of students who majored in engineering out of the total number of students surveyed.
We see that out of the 400 college seniors surveyed, 145 majored in engineering. Therefore, the proportion of students who majored in engineering is 145/400. To convert this proportion to a percentage, we multiply it by 100. So, (145/400) x 100 = 36.25%.
Therefore, 36.25% of the college seniors surveyed had an undergraduate major in engineering.
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For computer memory 1 MB = 210 bytes, 1 GB = 210 MB and 1TB = 210 GB. How many bytes are there in 1 TB? (MB is megabyte, GB is gigabyte, TB is terabyte)
Answer:
210
Step-by-step explanation:
Find the quotient of 12
5
÷ 2
5
.
1. Rewite the division as multiplication by the reciprocal.The rewritten expression is
2. Multiply the numerators to get , and then multiply the denominators to get
3. Simplify. The quotient is
the quotient of 125 ÷ 25.1 is approximately equal to 5/1.004 or 4.98 (rounded to two decimal places).
Why it is and what is quotient?
We can rewrite the division 125 ÷ 25.1 as multiplication by the reciprocal of 25.1:
125 ÷ 25.1 = 125 × 1/25.1
Now, we can simplify this expression by multiplying the numerators and denominators:
125 × 1/25.1 = (125/1) × (1/25.1) = 125/25.1
Finally, we can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 5:
125/25.1 = (25 × 5) / (5 × 5.02) = 5/1.004
Therefore, the quotient of 125 ÷ 25.1 is approximately equal to 5/1.004 or 4.98 (rounded to two decimal places).
In mathematics, the quotient refers to the result obtained from dividing one quantity by another. It is the answer to a division problem, and is typically expressed as a fraction or decimal. For example, the quotient of 10 divided by 2 is 5, written as 10 ÷ 2 = 5.
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A number is 12 more than another number. Twice the sum of two numbers is four. Find the two numbers
When a number is 12 more than another number and twice the sum of two numbers is four. Then two numbers are equals to the 7 and -5.
We have a number is 12 more than another number. Let the two numbers be 'a' and 'b' such that a> b. So, from the question, bigger number, a is 12 more than another number, b. In mathematics form, a = 12 + b --(1)
Also, twice the sum of two numbers is four, that is 2( a + b ) = 4 --(2)
We have to determine the values of two numbers. For this, solve the equation (1) and (2). Using the Substitution method,
Substitute, a = 12 + b from equation (1) to equation(2), 2( a + b ) = 4
=> 2( 12 + b + b) = 4
Simplify the expression
=> 24 + 4b = 4
=> 24 - 4 = -4b
=> 4b = - 20
dividing by 4 both sides
=> b = -5
Plugging the value of b = -5, in equation(1)
=> a = 12 + b = 12 + (-5)
=> a = 12 - 5 = 7
Hence required numbers are 7 and -5.
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Prove that for every pair of twin primes except for the pair (3,5) that the number between them is divisible by 6.”
Answer:
(3,5)
Step-by-step explanation:
Twin primes are a pair of prime numbers that differ by 2. Let's consider a pair of twin primes, p and p + 2 (where p > 3). We want to prove that the number between them, p + 1, is divisible by 6.
We know that every prime number greater than 3 can be written in the form 6k ± 1 for some integer k. This is because any integer can be written in one of six possible forms: 6k, 6k + 1, 6k + 2, 6k + 3, 6k + 4, or 6k + 5. However, we can eliminate the forms that are divisible by 2 or 3 (except for 2 and 3 themselves), leaving only 6k ± 1 and 6k ± 5. Since twin primes differ by 2, they must both be of the form 6k ± 1.
Let's consider p, the smaller of the twin primes. We know that p is of the form 6k ± 1 for some integer k. If p is of the form 6k + 1, then p + 2 is of the form 6k + 3, which is not prime (since it is divisible by 3). Therefore, p must be of the form 6k - 1. Then, p + 1 is of the form 6k, which means it is divisible by 2 and by 3.
Similarly, if p + 2 is of the form 6k - 1, then p is of the form 6k - 3, which is not prime (since it is divisible by 3). Therefore, p + 2 must be of the form 6k + 1. Then, p + 1 is of the form 6k, which means it is divisible by 2 and by 3.
Therefore, in either case, the number between the twin primes, p + 1, is divisible by 6. We have shown that this is true for any pair of twin primes except for the pair (3, 5).