The quadratic equation that has roots √3 + 1 / 2 and √3 - 1 / 2 and a leading coefficient of 9 is y = 9 · x² - 18√3 · x + 99 / 4.
How to find the quadratic equation associated with given real roots and leading coefficient
In this problem we must determine the quadratic equation such that its roots are √3 + 1 / 2 and √3 - 1 / 2 and whose leading coefficient is equal to 9. This can be found by substituting on factor form:
y = a · (x - r₁) · (x - r₂)
Where:
a - Leading coefficient.r₁, r₂ - RootsIf we know that r₁ = √3 + 1 / 2, r₂ = √3 - 1 / 2 and a = 9, then the quadratic equation is:
y = 9 · (x - √3 - 1 / 2) · (x - √3 + 1 / 2)
y = 9 · [x² - (√3 - 1 / 2) · x - (√3 + 1 / 2) · x + 3 - 1 / 4]
y = 9 · (x² - 2√3 · x + 11 / 4)
y = 9 · x² - 18√3 · x + 99 / 4
The quadratic equation is y = 9 · x² - 18√3 · x + 99 / 4.
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1
Suppose Vanessa can clean her bedroom in 30
minutes. Steven can clean the same room in only
15 minutes. If they work together, how long
will it take them to clean the bedroom ?
How long will it take them to clean the bedroom? 10 minutes
How to calculate the time of the work done together?
Given:
Vanessa (A)= 30 minutes
Steven (B) = 15 minutes
The total time taken if both work together is given by,
[tex]A+B=\frac{AB}{A+B} \\\\=\frac{30*15}{30+15}\\\\=\frac{450}{45}\\\\= 10[/tex]
So, it would take 10 minutes to clean the bedroom, if they work together
What is the time - work concept ?
Time and work is concerned with the amount of time it takes a person or a group of people to complete a task and the effectiveness of the work performed by each of them. The rate of work and the amount of time required are inversely proportional. Time taken = 1 / The rate of work.To learn more about time and work , refer:
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Simplify completely: cos/1−sin
There are basic six ratios in trigonometry that help in establishing a relationship between the ratio of sides of a right triangle with the angle. If θ is the angle in a right-angled triangle, formed between the base and hypotenuse, then
sin θ = Perpendicular/Hypotenuse
cos θ = Base/Hypotenuse
tan θ = Perpendicular/Base
The value of the other three functions: cot, sec, and cosec depend on tan, cos, and sin respectively as given below.
cot θ = 1/tan θ = Base/Perpendicular
sec θ = 1/cos θ = Hypotenuse/Base
cosec θ = 1/sin θ = Hypotenuse/Perpendicular
Given:
cos x/ 1- sin x
Using Trigonometric identities
1-sin²x = cos²x
So, on rationalizing
= (cos x) /(1-sinx) x (1+sinx)/(1+sinx)
= (cos x) (1+sinx) /(1²-sin²x)
= (cos x) (1+sinx) /(1-sin²x)
= (cos x)(1+sinx)/(cos²x)
= (1+sinx)/(cos x)
=1/ cos x + sin x / cos x
=sec x + tan x
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What is the exact value of tan 7pi/12? Thank you!
Remember that
[tex]tan(\frac{7\pi}{12})=tan(105^o)=-tan(75^o)=-tan(45^o+30^o)[/tex]and
[tex]tan(A+B)=\frac{tanA+tanB}{1-tanA*tanB}[/tex]therefore
[tex]tan(45^o+30^o)=\frac{tan45^o+tan30^o}{1-tan45^o*tan30^o}[/tex]substitute given values
[tex]tan(45^o+30^o)=\frac{1+\frac{\sqrt{3}}{3}}{1-(1)(\frac{\sqrt{3}}{3})}=\frac{\frac{3+\sqrt{3}}{3}}{\frac{3-\sqrt{3}}{3}}=\frac{3+\sqrt{3}}{3-\sqrt{3}}*\frac{3+\sqrt{3}}{3+\sqrt{3}}=\frac{9+6\sqrt{3}+3}{6}=\frac{12+6\sqrt{3}}{6}=2+\sqrt{3}[/tex]therefore
The answer is
[tex]tan(\frac{7\pi}{12})=-(2+\sqrt{3})[/tex]The answer is the second option
i only need the answer THANK YOU!
The trigonometry expression is cos Ф = 3√5/7 , sec Ф = 7/3√5, cosec Ф = 7/2, tan Ф = 2/3√5, cot Ф = 3√5/ 2
sin Ф = perpendicular(p) / hypotenuse(h)
= 2/7
base(b) = √(h²-p²)
=[tex]\sqrt{7^{2} - 2^{2} }[/tex]
= √45
= 3√5
cosФ = b/h
= 3√5/7
sec Ф = h/b
= 7/3√5
cosec Ф = h/ p
= 7/2
tan Ф = p/b
= 2/3√5
cot Ф = b/p
=3√5/ 2
Therefore cos Ф = 3√5/7, sec Ф = 7/3√5, cosec Ф = 7/2, tan Ф = 2/3√5, cot Ф = 3√5/ 2.
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Assume the random variable X is normally distributed with mean μ = 50 and standard deviation o=7. Find the 78th percentile
The 78th Percentile is 55.402
What is Percentile?
A Percentile (or centile) is a statistic measure that indicates the value below which a given percentage of observations in a group fall. The 20th percentile, for example, is the value (or score) below which 20% of the observations can be found.
The terms Percentile and Percentile rank are frequently used in the reporting of norm-referenced test scores. For example, a score in the 86th percentile, where 86 is the percentile rank, corresponds to a value below which 86% of the observations can be found. In contrast, a score in the 86th percentile indicates that it is at or below the value at which 86% of the observations can be found.
We know that,
μ = 50,
and σ = 7,
First, we need to find the z-score associated to this percentile.
We need to find the value [tex]Z_{p}[/tex] that solves the equation below.
[tex]Pr(Z < Z_{p}) = 0.78[/tex]
The value of [tex]Z_{p}[/tex] that solves the equation above cannot be made directly, it is solved either by looking at a standard normal distribution table or by approximation.
Based on this, we find that that the solution is [tex]Z_{p}[/tex] 0.772, because from the normal table we see that
Pr(Z<0.772)=0.78
Therefore, the percentile we are looking for is computed using the following formula:
[tex]P_{78}[/tex] = μ+ [tex]Z_{p}[/tex] × σ
= 50+0.772×7
= 55.405
Therefore, it is concluded that the corresponding 78-th percentile is found to be [tex]P_{78}[/tex] is 55.405.
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The diagonal of a rectangular filed 135m long is 157m. What is the Width of the field?
To solve this exercise let's make a sketch of the rectangular field first:
The length, width, and diagonal of the rectangular field form a right triangle, to determine the width of the rectangle you can apply the Pythagorean theorem.
This theorem states that the square of the hypotenuse is equal to the sum of the square of the legs of the triangle.
[tex]a^2+b^2=c^2[/tex]The diagonal is the hypotenuse of the triangle, the length is one of the legs "a" and the width is the other leg "b".
Write the theorem for b:
[tex]\begin{gathered} a^2-a^2+b^2=c^2-a^2 \\ b^2=c^2-a^2 \\ \sqrt[]{b^2}=\sqrt[]{c^2-a^2} \\ b=\sqrt[]{c^2-a^2} \end{gathered}[/tex]Replace the expression with c=157 and a=135
[tex]\begin{gathered} b=\sqrt[]{157^2-135^2} \\ b=\sqrt[]{24649-18225} \\ b=\sqrt[]{6424} \\ b=80.149\approx80.15 \end{gathered}[/tex]The width of the field is 80.15 meters.
12y=28x+609y=21x+45A.) no solution B.) many solutions C.) one solution
Step 1: Name the equation
12y = 28x + 60 ....... equation 1
9y = 21x + 45 ........... equation 2
Step 2: Eliminate one of the variables (either x or y ) in the two equations
To do this,
(12y = 28x + 60) multiply by 9
=> 108y = 252x + 540 ........ equation 3
(9y = 21x + 45 ) multiply by 12
=> 108y =252x + 540....... equation 4
step 4
equate equations 3 and 4
252x + 540 = 252x + 540
step 5 :solve the equation
252 x -252x = 540
0 = 0
Hence, the Equations has no solution
option A i
Three horse share five bales of hay equally. Five cows share eight bales of hay equally.
How much hay does each horse and each cow get?
Which animal gets more, a horse of a cow? How much more hay do they get?
How much hay does each horse and each cow get?
(a)The hay shared by 3 horses = 5 bales
So, the hay shared by 1 horse = x
Rewriting the statements ,
3 h = 5 b --(1)
1 h = x -(2)
By applying cross multiplication method to (1) and (2)
3 (x) = 5
x = 5/3
The hay shared by 1 horse = 5/3
(b)The hay shared by 5 cows = 8 bales
So, the hay shared by 1 cow = y
Rewriting the statements ,
5 c = 8 b --(3)
1 c = y -(4)
By applying cross multiplication method to (3) and (4)
5 (y) = 8
x = 8/5
The hay shared by 1 cow = 8/5
Which animal gets more, a horse of a cow?
Comparing the fractions,
The hay shared by 1 horse = 5/3 = 1.7
The hay shared by 1 cow = 8/5 = 1.6
1.7 > 1.6
So, a horse gets more than a cow
How much more hay do they get?
Considering the difference of fractions,
Amount of more hay consumed = Hay shared by a horse - hay shared by a cow.
[tex]=\frac{5}{3} -\frac{8}{5} \\\\=\frac{25-24}{15} \\\\=\frac{1}{15}[/tex]
The horse share [tex]\frac{1}{15}[/tex] more hay than cows
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Pls help I’m close finishing this assignment
A. No, Alexandra made a mistake going from the given expression to Step 1. Alexandra should have added 18 and 3 before evaluating the exponent. The order of operations says to perform operations inside parentheses before you evaluate exponents.
B. No, Alexandra made a mistake going from Step 1 to Step 2. Alexandra should have divided 9 by 9 before adding. The order of operations says to divide before you add.
C. Yes, Alexandra's work is correct.
The expression illustrates that C. Yes, Alexandra's work is correct.
What is an expression?The expression is the statement that illustrates that the variables given. In this case, two or more components are taken into consideration to describe the scenario.
The expression is given as:
= (18 + 3²) ÷ 3²
= (18 + 9) ÷ 9
= 27 ÷ 9
= 3
In this case, the work is correct.
Therefore, the correct option is C.
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1) Our baseball team last year had 10 boys. This year there are 15 boys. Find the percent of
change including whether it is an increase and decrease.
Answer: Hi, the answer would be 150% or in decimal form 1.5 this is an increase. There fore your answere is 1.5 which makes it an increase.
Find the maximum value of the function z = 4x+2y subject to the following constraints.
x≥5
y≥4
4x+3y≤56
The function at its maximum value will be at point (11,4) and its value is 52 which can be derived keeping in mind the constraints that have been given.
Maximizing a functionThe maximum value of any kind of a function is the place where the said function reaches its highest point, or vertex, on a graph. If your kind of quadratic equation has something as a negative of a term, it will also have a maximum value.
There are a bit of three ways to find that maximum, depending on which form of a quadratic function you have got.
Herein we can find the closed region ABC from the three equations or constraints given
and can mark the points as
A = (5,4)
B = (5, 12)
C = (11, 4)
if we put up the values of the points, we get
at (11, 4), Z = 4 x 11 + 2 x 4
=52 that is the maximum value amongst the three.
Thus, the maximum point of the function can be calculated as (11,4)
How do you extract out the maximum or minimum value of a function?We will at first set out the first derivative of the function to absolute zero and then solve for x to get the exact critical point.
If we then take the second derivative or f’’(x), there in we can find out whether this point will be a maximum or minimum. If the second derivative is seen as positive, it will be a minimum value
What is the maximum value of a function called?The maximum or minimum value that is calculated over the entire function is called an “Absolute” or “Global” maximum or minimum of the function.
There is presence of only one global maximum (and only one global minimum) but there can be several that is more than one local maximum or minimum
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If Volume = 216 cm^3, what is the length of one edge of Julie's jewelry box?
The volume of a cube = Side³
The length of one edge of Julie's Jewelry box is 6 cm.
What is a cube?A cube is a solid three-dimensional figure, which has 6 square faces, 8 vertices, and 12 edges.
The volume of a cube = Side³
We have,
The volume of the Jewelry box = 216 cm³
The length of one edge of the box = L
The volume of a cube is given as:
V = L³
216 = L³
[ 6 x 6 x 6 = 216 ]
L = ∛216
L = ∛6³
L = 6
Thus,
The length of one edge of Julie's Jewelry box is 6 cm.
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Use the vertex (0,3) and a point on the graph (2,7) to find the general form of the equation of the quadratic function.
The line which is passing through the point on the graph (2,7) and with vertex (0,3) having equation is [tex]x^{2}[/tex] -y +3 = 0
The polynomial equations of degree two in one variable of type f(x) = ax^2 + bx + c = 0 and with a, b, c are R and a [tex]\neq[/tex] 0, are known as quadratic equations. It is a quadratic equation in its general form, where "a" stands for the leading coefficient and "c" for the absolute term of f(x).
The general equation of the quadratic equation if (h, k) is the vertex and (x, y) is a point on the curve is given by
y = [tex]a(x-h)^{2} + k[/tex] ......eqn 1
In the situation where a must be determined at position (h, k) and (x, y)
In the above question, it is given that,
vertex is (h,k) = (0,3) and the point on the graph is (x,y) = (2,7)
Now we'll put the points in the general equation of the quadratic equation to get the desired equation
y = [tex]a(x-h)^{2} + k[/tex]
y = [tex]a(x-0)^{2}[/tex] + 3 .....eqn 1
Now putting the value of (x,y)
7 = [tex]a(2-0)^{2}[/tex] + 3
7 = 4a + 3
4a = 4
a = 1
putting the value of a in the equation 1, we get
y = [tex]1(x-0)^{2}[/tex] + 3
y = [tex]x^{2}[/tex] + 3
[tex]x^{2}[/tex] -y +3 = 0
Hence, the equation of the line passing through a point on the graph (2,7) with the vertex (0,3) is [tex]x^{2}[/tex] -y +3 = 0
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Find the 11th term of the sequence: 32805, 10935, 3645...
first term a=32805
common ration: r=10935/32805=1/3
11th term:
[tex]\begin{gathered} a_{11}=32805(\frac{1}{3})^{11-1} \\ a_{11}=32805(\frac{1}{3})^{10} \\ a_{11}=32805(\frac{1}{59049}) \\ a_{11}=0.555 \end{gathered}[/tex]So the 11th term is 0.555
Write a paragraph proof of your conclusion in part A. To begin your proof, draw radii OA and OC.
In the given figure we have that
AC=AD -----> by the radius of the circle
Δ ACO -------> is a right triangle
Δ ADO -------> is a right triangle
Applying the Pythagorean Theorem in both triangles
AC^2=CO^2+AO^2
AD^2=DO^2+AO^2
Remember that
AC=AD ----> radius of the circle
so
CO=DO
Δ ACO≅Δ ADO -----> by SSS
therefore
is proved that the radius that bisects the chord is a perpendicular bisector
In right triangle ABC, CD¯¯¯¯¯ is the altitude to hypotenuse AB¯¯¯¯¯. If AB = 28.3¯ and DB = 12, find CD.
The side of the right triangle CD is 25.63 units
How to find side of a right triangle?A right triangle is a triangle that has one of its angles as 90 degrees.
Base on the angles, the side of a right triangle can be named as follows:
opposite sidehypotenuse sideadjacent sideTherefore, the side CD of the right triangle can be found as follows:
using Pythagoras theorem,
c² = a² + b²
Hence,
CD = √28.3² - 12²
CD = √800.89 - 144
CD = √656.89
CD = 25.6298653918
Therefore,
CD = 25.63 units
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Say a certain manufacturing industry has 63.1 thousand jobs in 2008, but is expected to decline at arate of 1.7 thousand jobs per year from 2008 to 2018. Assuming this holds true, what will be this induchange from 2008 to 2018?a. 70%b. -27%C. -17%d. -75%
The inicial number of jobs is 63.1 thousand, and each year this number will decrease by 1.7 thousand.
So the change from 2008 to 2018, that is, 10 years, will be:
[tex]10\cdot(-1.7)=-17[/tex]To find the change in percentage, we need to divide the change (-17) by the inicial amount of jobs (63.1), then we have that:
[tex]\frac{-17}{63.1}=-0.27=-27\text{\%}[/tex]So the change in this period is -27%, therefore the answer is B.
Answer the image below
The value of the function f'(g(x)) is -10
What is a function?
an expression or rule or law that describes a relationship between one variable and another variable. i.e. between independent variable and dependent variable is known as function.
We are given f(-6)=-3, f'(-6)=-5, g(2)=-6, g'(2)=2
We are asked to find F'(2) where F(x)= f(g(x))
We will use the algebra of function to solve the given problem and find the required value
We have
[tex]F(x)=f(g(x))\\F'(x)= f'(g(x))*g'(x)[/tex]
Substituting x=2 in the above equation we get
[tex]F'(2)=f'(g(2))*g'(2)\\F'(2)=f'(-6)*2\\F'(2)=-5*2\\F'(2)=-10[/tex]
Hence the value of F'(2)= -10 as asked
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Which of the following is the correct mathematical expression for:
Halve a number and then increase it by five
Answer:
1/2 x + 5
Step-by-step explanation:
Haf of x = 1/2 x
then add 5 : 1/2 x + 5
each card in a set of cards has a different number of 1 to 12 written on it a student randomly chooses a car record two numbers shown on the card and replaces the card the table shows the results of the students choosing a card 25 x number of Cars 1 2 3 4 5 6 7 8 9 10 11 12 number of outcomes +23-142-311-2213 space on the data in the table how many of the next 125 cars of the student chooses could be expected to have a number shown on a car that is a multiple of 4
Given data:
The given table is shown.
The expression for the probability that the number is multiple of 4.
[tex]\begin{gathered} P(4m)=\frac{8}{25} \\ =0.32 \end{gathered}[/tex]When 125 times the cards are chosen then expected number card is multiple of 4 is,
[tex]\begin{gathered} E(4m)=125\times0.32 \\ =40 \end{gathered}[/tex]Thus, the correct option is (A).
Answer the questions below.
(a) Daily high temperature
The daily high temperature does not depend on the number of ticket sales.(b) Amount of sleep
The amount of medication influences the amount of sleep.(c) Input U
The independent variable is the input.66 randomly selected students were asked the number of pairs of shoes they have. Let X representthe number of pairs of shoes. The results are as follows:# of Pairs of Shoes 4Frequency 75 68 9 10 114 11 11 11 2 12 8Round all your answers to 4 decimal places where possible.The mean is:The median is:The sample standard deviation is:The first quartile is:The third quartile is:What percent of the respondents have at least 10 pairs of shoes?9617% of all respondents have fewer than how many pairs of shoes?
Given data
1. The mean
[tex]\frac{\sum fx}{\sum f}=\frac{505}{66}=7.6515[/tex]2. The median
[tex]\begin{gathered} \text{ Median = (}\frac{\text{N+1}}{2})th \\ =(\frac{66+1}{2})th=(\frac{67}{2})th=33.5th \\ \text{The median is the 33.5th }term\text{ which is between 7 and 8} \\ \text{The median is }\frac{\text{7+8}}{2}=\frac{15}{2}=7.5 \end{gathered}[/tex]3. The sample standard deviation
[tex]\begin{gathered} SD\text{ =}\sqrt[]{\frac{\sum f|x-\bar{x}|}{\sum f}} \\ \bar{x}=7.6515 \\ SD=\sqrt[]{4.8028} \\ SD=2.1915 \end{gathered}[/tex]If it is 3:40 p.m. in Denver, what time is it in Miami?
Answer:
5:40 P.M.
Explanation:
From the time map:
• When the time in Denver is 10.00 AM
,• The time in Miami is 12:00 Noon
This means that Miami is 2 hours ahead of Denver.
Thus, if it is 3:40 p.m. in Denver:
[tex]\begin{gathered} \text{Time in Miami}=3\colon40+2\colon00 \\ =5\colon40p.m\text{.} \end{gathered}[/tex]It is 5:40 P.M. in Miami.
what is 2×700 please tell me
2 multiply by 700 is 1400
An easier trick to multipy numbers with zeros is;
Remove the zeros first, multiply the numbers, then add back the zeros
That is;
2 x 7 = 14
Then add back the zeros
That is;
1400
Solve for this angle.
X+87° 2x° i have to solve for the angle with the 2 on it
Answer:
x=31
Step-by-step explanation:
x+87+2x=180(angles on a straight line is equal to 180)
3x+87=180
3x=180-87
3x=93
x=31
please mark brainliest
In the diagram below, AB || DEF, AE and BD intersect at C, m2B = 43°, and mZCEF = 152°.
From the picture, we have the necessary information:
The supplementary angles.
The parallels being intersected by two secant lines. And alternate interior angles are congruent under this condition.
The other angle of the triangle is then:
As a result, the angles vertical angles, therefore
< BAC = 28 (it confirms the alternate interior angle Therefore, we can write all the angles in the following drawing:
Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. 7t+5−−−−−√=7 Enter the exact answers.
The given equation is
[tex]\sqrt[]{7t+5}=7[/tex]Square both sides to cancel the square root
[tex]\begin{gathered} (\sqrt[]{7t+5})^2=(7)^2 \\ 7t+5=49 \end{gathered}[/tex]Subtract 5 from both sides
[tex]\begin{gathered} 7t+5-5=49-5 \\ 7t=44 \end{gathered}[/tex]Divide both sides by 7
[tex]\begin{gathered} \frac{7t}{7}=\frac{44}{7} \\ t=\frac{44}{7} \\ t=6\frac{2}{7} \end{gathered}[/tex]The solution is t = 44/7 OR t = 6 2/7 (mixed number)
The area of the front yard for question number 1 is 60x^2 + 12x - 21 The grass area of the backyard for question number 2 is 81x^2 - 212x + 68
Given,
The area of the front yard is 60x^2 + 12x - 21 .
The grass area of the backyard is 81x^2 - 212x + 68.
The total are is the sum of the front yard and the back yard.
So, to obtain the total area adding the area of the front yard and back yard.
[tex]\begin{gathered} \text{Total area= area of front yard+area of back yard} \\ =60x^2+12x-21+81x^2-212x+68 \\ =60x^2+81x^2+12x-212x-21+68 \\ =x^2(60+81)+x(12-212)-21+68 \\ =141x^2-200x+47 \end{gathered}[/tex]Hence, the total area is 141x^2-200x+47.
Find 75% of 150. Round to the nearest tenth if necessary.
Find 75% of 150
First, let's convert 75% to decimal by dividing by 100:
[tex]\frac{75}{100}=0.75[/tex]Now, we will multiply 0.75 with 150 to find out answer:
[tex]\begin{gathered} 0.75\times150 \\ =112.5 \end{gathered}[/tex]Answer112.5I believe that it would be 112.5.
__________________________________________
Convert 75% to a decimal by dividing it by 100:
[tex]\frac{75}{100} = 0.75[/tex]
Next, multiply 0.75 to 150:
[tex]0.75[/tex] × [tex]150 = 112.5[/tex] ← Final Solution
Hope this helps!
Four people ordered total bill for soup and four people ordered stew is $31 .If only 2 people had ordered soup and one person ordered stew ,the bill would have been $12.25 .How much is each order of soup and stew ?
Let price of 1 soup be "x" and 1 stew be "y".
4 soup and 4 stew is $31, thus we can write:
[tex]4x+4y=31[/tex]Also, given
2 soup and 1 stew costs $12.25, thus we can write:
[tex]2x+y=12.25[/tex]Let' solve this for y and substitute it back to Equation 1. Then we will solve for x. The process is shown below:
[tex]y=12.25-2x[/tex]Substituting,
[tex]\begin{gathered} 4x+4y=31 \\ 4x+4(12.25-2x)=31 \\ 4x+49-8x=31 \\ 49-31=8x-4x \\ 18=4x \\ x=\frac{18}{4} \\ x=4.5 \end{gathered}[/tex]Finding y:
[tex]\begin{gathered} y=12.25-2x \\ y=12.25-2(4.5) \\ y=3.25 \end{gathered}[/tex]Thus,
Answer:
Each order of Soup = $4.50
Each order of Stew = $3.25