(a) The values of the real constants λ and μ are -6 and 8, respectively.
(b) The roots of the equation are (1 + i), (1 - i), and -4.
We are given an equation. The equation is cubic in nature. It means the equation has three roots. The equation is given below.
x³ + 2x² + λx + μ = 0
One of the roots of the equation is given to us. The known root is "1 + i". The given root is a complex root. We know that complex roots occur in conjugate pairs. So, another root of the equation is "1 - i". Let the third root be represented by the variable "a". We can form the factors as given below.
[x - (1 + i)][x - (1 - i)](x - a) = 0
[x² - x(1 - i) - x(1 + i) + (1 + i)(1 - i)](x - a) = 0
[x² - x + ix - x - ix + (1² - i²)](x - a) = 0
[x² - 2x + (1 + 1)](x - a) = 0
(x² - 2x + 2)(x - a) = 0
x³ - ax² - 2x² + 2ax + 2x - 2a = 0
x³ - (a + 2)x² + (2a + 2)x - 2a = 0
This equation must be the same as the given equation. Now we will compare the coefficients of different powers of "x".
First, we compare the coefficient of x².
2 = -(a + 2)
a + 2 = -2
a = -4
Now, we compare the coefficient of x.
λ = 2a + 2
λ = 2*(-4) + 2
λ = -8 + 2
λ = -6
Now we compare the constant terms.
μ = -2a
μ = -2*(-4)
μ = 8
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Given 3 and one-tenth times negative 6 times seven-twelfths, determine the product.
eighteen and one sixtieth
negative eighteen and 7 over 120
ten and 17 over 20
negative 10 and 17 over 20
The given statement expression is equivalent to -2 [tex]\frac{1}{10}[/tex].
What is a expression? What is a mathematical equation? What is Equation Modelling? What are complex numbers?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem. In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted [i], called the imaginary unit and satisfying the equation -
[tex]{\displaystyle i^{2}=-1}[/tex]
Every complex number can be expressed in the form a + bi, where [a] and [b] are real numbers
We have the following statement -
3 and one-tenth times negative 6 times seven-twelfths
We can write -
3 (1/10 x - 6) × 7/12
- 3 [tex]\frac{3}{5}[/tex] × 7/12
-18/5 × 7/12
-3/5 x 7/2
- 21/10
- 2 [tex]\frac{1}{10}[/tex]
Therefore, the given statement expression is equivalent to -2 [tex]\frac{1}{10}[/tex].
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Which equation represents a proportional relationship?
OA. y = 4x
OB. y=3x2
OC. y = 5x+2
OD. y = 7
A is the equation representing proportional relationship .
What is proportional relationship ?Relationships between two variables that are proportional occur when their ratios are equal. Another way to consider them is that in a proportionate relationship, one variable is consistently equal to the other's constant value. The "constant of proportionality" is the name of this constant.
CalculationIf the y intercept is anything but 0, then the linear function is not proportional. So C is not proportional (2 is the y intercept)
B is a quadratic and it will not give proportional results. y/x is not a constant.
D has no x value and is not going to give a proportional result. y/x is undefined.
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A regular pentagon is inscribed in a circle as shown. 1. Find the measure of minor arc cut off by one of the diagonals.2. Find the length of the same minor arc in problem 16a, given the radius of the circle is 10 cm. Leave the answer is terms of pi.
SOLUTION:
Step 1:
we are to find the measure of minor arc cut off by one of the diagonals;
The sum of interior angles in a pentagon is:
[tex]\begin{gathered} (n-2)\text{ x 180} \\ (5-2)\text{ x 180} \\ 3\text{ x 180} \\ 540 \end{gathered}[/tex]Each interior angle of a regular pentagon is
[tex]\frac{540}{5}\text{ = 108}[/tex]So the size of the major arc can be gotten by the circle theorem; the angle at the centre is twice the angle at the circumference.
[tex]\text{The major arc = 2 x 108 = 216}[/tex]Then recall,
[tex]\begin{gathered} \text{The major arc + the minor arc = 360 (sum of angles at a point)} \\ 216\text{ + the minor arc = 360} \\ \text{The minor arc = 360 - 216} \\ \text{The minor arc =144} \end{gathered}[/tex]Step 2:
We are to find the length of the same minor arc in problem;
[tex]\frac{\theta}{360}\text{ x 2 }\pi\text{ r}[/tex]Where our angle (titan) is 144 and radius is 10
[tex]\begin{gathered} \frac{144}{360}\text{ x 2 x }\pi\text{ x 10} \\ \\ 8\pi \end{gathered}[/tex]So the length of the minor arc, given that the radius is 10 cm is 8 pi
√20 - 2√5solve with using adding and subtracting radicals
the initial expression is:
[tex]\sqrt[]{20}-2\cdot\sqrt[]{5}[/tex]so to rest them we want to have the same radical so we can rewrite the root of 20 so:
[tex]\begin{gathered} \sqrt[]{4\cdot5}-2\cdot\sqrt[]{5} \\ 2\cdot\sqrt[]{5}-2\cdot\sqrt[]{5}=0 \end{gathered}[/tex]write the new coordinates after a glide reflection described by. (x,y) (x, y-2) reflection in x=0.
To find the new coordinates, we just have to use the given rule
[tex](x,y)\rightarrow(x,y-2)[/tex]Let's apply it to each vertex
[tex]\begin{gathered} A(2,10)\rightarrow A^{\prime}(2,10-2)\rightarrow A^{\prime}(2,8) \\ B(3,-4)\rightarrow B^{\prime}(3,-4-2)\rightarrow B^{\prime}(3,-6) \\ C(-3,1)\rightarrow C^{\prime}(-3,1-2)\rightarrow C^{\prime}(-3,-1) \end{gathered}[/tex]Therefore, the new coordinates are A'(2,8), B'(3,-6), and C'(-3,-1).15. Which of the following relations defines y as a function of x?
Remember that
The data set is a function if every element of the domain corresponds to exactly one element of the range
so
Verify each option
I ------> is a function (every element of the domain corresponds to exactly one element of the range)
II -----> is not a function (every element of the domain does not correspond to exactly one element of the range)
III ---> is a function (every element of the domain corresponds to exactly one element of the range)
IV ----> is not a function (every element of the domain does not correspond to exactly one element of the range)
therefore
The answer is
Both I and III define a y as function of xWhat number will complete the point-slope equation that models this scenario?
y−50,000=[_______](x−10)
−3,000
−2,000
2,000
3,000
A Point-Slope equation of a straight line models this scenario: y -50,000 = -3,000(x -10).
What is the Point-slope form?The equation of the straight line has its slope and given point.
If we have a non-vertical line that passes through any point(x1, y1) has gradient m. then general point (x, y) must satisfy the equation
y-y₁ = m(x-x₁)
We can recognize any points of the graph.
(0,80000) ; (10,50000)
Given that;
y -50,000 =______(x -10)
We need to find the slope.
Slope = Change in y / Change in x
Slope = 80,000 - 50,000 / 0 - 10 = 30,000 / -10
= -3,000
The slope is equal to -3000 that is Choice A; y -50,000 = -3,000(x -10)
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The complete question is;
The newspaper in Haventown had a circulation of 80,000 papers in the year 2000. In 2010, the circulation was 50,000. With x = 0 representing the year 2000, the graph below models this scenario.
LaDawn needed wire for her Halloween wreath. How many 1/4 inch pieces can be cut from a piece of wire that is 5/8 inches long?
The length of each piece of wire is 2.5 inches.
What are fractions?Fractions are used to depict the components of a whole or group of items. Two components make up a fraction. The numerator is the number that appears at the top of the line. It specifies how many identically sized pieces of the entire event or collection were collected. The denominator is the quantity listed below the line. The total number of identical objects in a collection or the total number of equal sections that the whole is divided into are both displayed. A fraction can be expressed in one of three different ways: as a fraction, a percentage, or a decimal. The first and most popular way to express a fraction is in the form of the letter ab. Here, a and b are referred to as the numerator and denominator, respectively.
Let the number of pieces = x
The length of the wire = 5/8 inches
The length of one piece of wire = 1/4 inch
According to the question,
1/4 x = 5/8
So, x = 5/8 × 4
x = 20/ 8
x = 5/2
x = 2.5 inch
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You and your best friend have $10,000 each, which you would like to keep in a savings account. 5/3rd bank offers 4.23% continuous interest and bank of america will give 4.4% interest compounded quarterly. what would be the difference between the two investments after 35yrs? (round to the nearest dollar (whole number))
The difference between the two investments after 35yrs is $4700 and Bank of America is offering higher interest than 5/3rd banks.
Explain compound interest and continuous interest.Most interest is compounded semiannually, quarterly, or monthly. Continuous compounding assumes that interest is compounded indefinitely and reentered on the balance sheet. The continuous compounding formula takes into account four variables.
For the given question,
5/3rd bank offers continuous interest:
For continuous interest, A = P[tex]e^{rt}[/tex]
where,
P = the initial amount
A = the final amount
r = the rate of interest
t = time
e is a mathematical constant where e ≈ 2.7183.
A = 10000 × [tex]2.71^{0.042*35}[/tex]
A = 10000 × 4.04
A = $40400 approximately
Interest = Amount - Principle
Interest = 40400 - 10,000
Interest = $30400
Bank of America offers compound interest:
For compound interest:
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
A = 10000 × 4.51
A = $45100
Interest = Amount - Principle
Interest = 45100 - 10,000
Interest = $35100
The above calculation explains Bank of America is offering higher interest than 5/3rd bank by $35100 - $30400 = $4700
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Graph each system and determine the number of solutions that it has. If there is no solution, type NS. Ifit has a solution, state it. When entering solutions, enter them as a coordinate pair. Do not includespaces in answers.3x +y = -33x +y = 3Number of Solutions:Solution:
we have the system
3x +y = -3
3x +y = 3
In this problem we have two different parallel lines
Remember that the solution of the system is the intersection point both lines
In this problem the lines don't intersect (are parallel)
therefore
the system has no solution NS
isolate the variable y in the first equation
y=-3x-3 -----> equation 1
isolate the variable y in the second equation
y=-3x+3 -----> equation 2
Solve the system of equations
equate equation 1 and equation 2
-3x-3=-3x+3
solve for x
-3x+3x=3+3
0=6 ------> is not true
that means
the system has no solution (NS)
suppose that there are two types of tickets: advanced and same day. advance tickets cost $25 and same day tickets are $30.for one performance there were 65 tickets sold in all. and the total amount paid for them was $1800.how many tickect of each were sold
The number of advance and same day tickets sold are 25 and 35 respectively.
How many tickect of each were sold?let
number of advance tickets sold = xNumber of same day tickets sold = yx + y = 65
25x + 30y = 1800
From (1)
x = 65 - y
Substitute x = 65 - y into (2)
25x + 30y = 1800
25(65 - y) + 30y = 1800
1625 - 25y + 30y = 1800
5y = 1800 - 1625
5y = 175
y = 175/5
y = 35
Substitute y = 35 into (1)
x + y = 65
x + 35 = 65
x = 65 - 35
x = 25
Therefore, Simultaneous equation is used to determine the number of advance and same day tickets sold.
25 advance tickets and 35 same day tickets were sold.
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prime or composite
number 1
67
67 is a prime number and 1 is neither prime nor composite number.
What is prime number and composite number?
A prime number is a number which has exactly two factors i.e. '1' and the number itself. A composite number has more than two factors, which means apart from getting divided by 1 and the number itself, it can also be divided by at least one positive integer.
The given two numbers are 1 and 67.
Since, the prime numbers from 1 to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
67 has exactly two factors i.e. 1 and the number 67 itself.
Therefore, 67 is a prime number.
Due to the fact that it only has one factor, 1, number is not a prime.
Hence, 67 is a prime number and 1 is neither prime number nor composite number.
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A store that sells cell phones has cell-phone covers in a variety of 6 colors and 6 designs. An employee will randomly choose a cell-phone cover to give to each customer who purchases a new cell
phone. Which simulation best models the outcomes of the possible cell-phone covers?
rolling two number cubes at the same time and recording the results
flipping a coin twice and recording the results
rolling a number cube and recording the results
rolling a number cube, flipping a coin, and recording the results
Answer: I am pretty sure that the answer is flipping a coin twice and recording the result
Step-by-step explanation: my reasoning is you can compare the results unlike the rest
Describe how a drywaller might use a formula to estimate the
number of sheets of drywall needed for a room.
Answer:
Square footage
Step-by-step explanation:
What is an equation of the line that passes through the points (6,0) and (3, 4)?
The slope-intercept form of the equation of a straight line is given to be:
[tex]y=mx+b[/tex]where m is the slope and b is the intercept on the y-axis.
The slope is calculated using two points on the line by the formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Therefore, for this line, the slope will be:
[tex]m=\frac{4-0}{3-6}=-\frac{4}{3}[/tex]Therefore, this value for the slope is substituted back into the equation for the line:
[tex]y=-\frac{4}{3}x+b[/tex]At the point (6, 0), we can calculate the value of b to be:
[tex]\begin{gathered} 0=-\frac{4}{3}(6)+b \\ b=\frac{4}{3}(6) \\ b=8 \end{gathered}[/tex]Therefore, the equation is:
[tex]y=-\frac{4}{3}x+8[/tex]Write 280,000,000 in scientific notation.
0.28 × 109
28 × 108
2.8 × 108
2.8 × 109
Answer:
= 4.875 × 105
(scientific notation)
= 4.875e5
(scientific e notation)
= 487.5 × 103
(engineering notation)
(thousand; prefix kilo- (k))
help meeeeeeeeeeeeeee pleaseeeeeee
Answer: Width = 4.7 meters, Length = 6.7 meters
Step-by-step explanation:
Let the width be w. Then, the length is w+2.
[tex]w(w+2)=32\\\\w^2 +2w=32\\\\w^2 +2w-32=0\\ \\ w=\frac{-2 \pm \sqrt{2^2 -4(1)(-32)}}{2(1)}\\\\w \approx 4.7\\\\\implies w+2 \approx 6.7[/tex]
Of a sample of 500 students surveyed, 300 students say they listen to music when they study. In a school with 1500 students, how many students would be likely to listen to music while studying? hi 300 students 500 students 900 students 1200 students
Given:
Surveyed Sample = 500 students
Number of students who listen to music = 300
The probability of students that listens to music is given by the formula:
[tex]pr\text{ =}\frac{number\text{ of students that listen to music}}{Total\text{ number of student}}\text{ =}\frac{300}{500}[/tex][tex]pr\text{ = }\frac{3}{5}[/tex]In a sample of 1500 students, to find the probable number of students that will listen to music
we will multiply the probability we obtained initially by the sample space
=>
[tex]\begin{gathered} \frac{3}{5}\text{ x 1500} \\ \\ \frac{4500}{5} \\ \\ =900 \end{gathered}[/tex]It is likely that 900 students will listen to music while studying
Jasper opened a savings account and deposited 200.00 the account earns 9%interest compounded annually if he wants to use the money to buy a new bicycle in 2 years how much will he able to spend on the bike
The formula for calculating the amount of money after compounding for a period of time is expressed as;
A = P(1+r/n)^nt
P is the principal (amount deposited)
r is the rate
t is the time
n is the time of compounding
Given
P = 200.00
r = 9% = 0.09
t = 2 years
n = 1 year
Substitute the given parameters into the formula
A = 200(1+0.09/1)^1(2)
A = 200(1.09)^2
A = 200(1.1881)
A = 237.62
Hence he will be able to spend 237.62 on the bike
true or false? The sun of an even whole number and an odd whole number is always odd.
Answer:
the sum of an even whole number and odd whole number is always odd ,
True
What percent of 120 is 180%
What percent of 120 is 180
we know that
120 represent the 100%
so
Applying proportion
100/120=x/180
solve for x
x=(100/120)*180
x=150%
the answer is 150%Do 5 + 3 and 3 + 5 represent the same express? Explain.
5 + 3 = 8
3 + 5 = 8
The represent the same expression
|x - 7| ≥ 1?
what is the solution?
Answer: Inequality form: x is less then or equal to 6 or xis greater then or equal to 8
Interval Notation: (-infinity,6] u [8,infinity)
Answer:
x ≤ 6 ∨ x ≥ 8
Step-by-step explanation:
x-7 ≥ 0
when
x≥7
if x≥7
x-7≥ 1
x ≥ 7 + 1
x ≥ 8
if x < 7
-x + 7 ≥ 1
-x ≥ -7 + 1
-x ≥ -6
x ≤ 6
Final solution
x≤ 6 ∨ x ≥ 8
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Libby wants to draw a scale diagram of the Sun and the Earth.
The diameter of the Sun is 1.4 million km to 2 s.f.
The diameter of the Earth is 13,000 km to 2 s.f.
k to task
Libby decides to draw the Earth with a diameter of 2 cm.
Work out the diameter of the Sun that she needs to draw.
Give your answer in metres to 2 significant figures.
Watch video
The diameter of the Sun that she needs to draw, considering the scale, is of:
2.15 m.
What is a scale?The meaning of the scale of a drawing is how much each unit in the drawing represents in actual distance.
In this problem, the actual diameters are given as follows:
Sun: 1,400,000 km.Earth: 13,000 km.The drawn diameters are given as follows:
Sun: x cm.Earth: 2 cm.From the scale of the earth, the number of km represented with each cm is given as follows:
13000/2 = 6500 km represented by each cm.
(division of the actual distance by the scale).
Hence the number of cm that is needed to represent 1,400,000 km is of:
1400000/6500 = 215 cm.
(division of the total distance by the scale).
Each cm is composed by 100, the diameter of the Sun in meters is of:
215/100 = 2.15 m.
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If point (x, y) is reflected over the line y = x, the resulting point is (y,x).TrueFalse
When we reflect a point over y = x, the x and y coordinates become swapped and switch places, that is why (x,y) results to (y,x).
Therefore, the answer is true.
In 8 days, the temperature dropped 12 degrees. How many degrees per day did the temperature drop
Answer:
x = 1.5°
Step-by-step explanation:
8 days = 12°
1 day = x
cross-multiply
8x = 12
Divide both sides by the coefficient of x, which is 8
8x/8 = 12/8
x = 1.5°
Hence, 1 day = 1.5°
a line that passes through the point (2,1) and is parallel to the line y = -2x + 5
A line in slope-intercept form is represented by the following equation:
[tex]\begin{gathered} y=mx+b \\ \text{where,} \\ m=\text{slope} \\ b=y-\text{intercept} \end{gathered}[/tex]Parallel lines have the same slope, then if the line is parallel to the line y=-2x+5, the slope of the parallel line would be -2 too.
Now, by the given point (2,1) and knowing the slope of the line, we can use the slope-point form of the line equation to get the slope-intercept form:
[tex]\begin{gathered} y-y_0=m(x-x_0) \\ \text{where,} \\ (x_0,y_0)\text{ is the given point} \\ m\text{ is the slope} \end{gathered}[/tex]Substituting:
[tex]\begin{gathered} y-1=-2(x-2) \\ y=-2x+4+1 \\ y=-2x+5 \end{gathered}[/tex]Notice that the parallel line would be the same original line, y=-2x+5.
What values of c and d would make
the following expression represent a real number?
i(2 + 3i)(c + di)
c = 2, d = 3
c = –2, d = 3
c = 3, d = –2
c = –3, d = –2
The values of c and d that would make that the given expression represent a real number are c = –3, d = –2.
What is meant by the term real number?In mathematics, a real number is a quantity that may be articulated as just an infinite decimal expansion. In comparison to the natural numbers 1, 2, 3,..., which are derived from counting, real numbers are employed in measurements of varying different quantities including such size and time.For the given question;
The expression is given as;
= i(2 + 3i)(c + di)
Simplify the expression;
= i(2c + 2di + 3ci + 3di²)
= 2ci + 2di² + 2ci² + 3di³
= (2c - 3d)i - 2d - 3c
In order is for expression to yield a real number, the coefficient of must now be zero.
Check the values from option;
2c - 3d = 2 x 2 - 3x 3 = -5
2c - 3d = 2 x -2 - 3 x -3 = -13
2c - 3d = 2 x 3 + 3 x -2 = -1
2c - 3d = 2 x -3 -3 x - 2 = 0
Thus, the correct values for the expression to becomes real number is
c = –3, d = –2.
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Set up a proportion and soive for
side TV (y):
ATUV-AXWV
As we Know that, ΔTUV[tex]\sim[/tex] ΔXWV
So, for the given proportion TV(y) = 21.
What is Proportion?
A proportion is an equation that sets two ratios at the same value. For instance, you could express the ratio as follows: 1: 3 if there is 1 boy and 3 girls (for every one boy there are 3 girls) There are 1 in 4 boys and 3 in 4 girls. 0.25 are male (by dividing 1 by 4)
What is Ratio?
The link between two or more things in terms of quantity, amount, or size.
Given that,
ΔTUV[tex]\sim[/tex] ΔXWV
∠T = ∠X
∠U = ∠W
∠TUV = ∠XWV
So, TU/WX = TV/WV
10/10 = TV/21
1 = TV/21
TV = 21
So, for the given proportion the value for the side TV(y) = 21
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Sketch the graph of y= sin x° for 0≤x≤360
Answer:
Step-by-step explanation:
This is a sketch graph of y= sin x° for 0 ≤ x ≤360.
It is called as Sine curve. A mathematical curve described in terms of the sine trigonometric function, of which it is the graph, is known as a sine wave, sinusoidal wave, or simply sinusoid. It is a smooth periodic function and a particular kind of continuous wave.
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