Answer:
k = -12.6
Step-by-step explanation:
Given rational polynomial:
[tex]\dfrac{x^3-2x^2+kx-12}{x-5}[/tex]
When we divide a polynomial f(x) by (x − d) the remainder is f(d).
If the remainder is zero, then (x - d) must be a factor of f(x).
If a cubic polynomial is divided by (x - d) then:
[tex]\implies (x-d)(ax^2+bx+c)[/tex]
where:
a is the leading coefficient-cd is the constantIf the constant of the cubic polynomial is -12 and d = 5:
[tex]\implies -cd=-12[/tex]
[tex]\implies -5c=-12[/tex]
[tex]\implies c=2.4[/tex]
The leading coefficient of the cubic polynomial is one. Therefore:
[tex]\implies (x-5)(x^2+bx+2.4)[/tex]
Expand and collect like terms:
[tex]\implies x^3+bx^2+2.4x-5x^2-5bx-12[/tex]
[tex]\implies x^3+(b-5)x^2+(2.4-5b)x-12[/tex]
The find the value of b, compare the coefficients of the term in x²:
[tex]\implies b-5=-2[/tex]
[tex]\implies b=3[/tex]
To find k, substitute the found value of b into the coefficient of x:
[tex]\implies k=2.4-5b[/tex]
[tex]\implies k=2.4-5(3)[/tex]
[tex]\implies k=-12.6[/tex]
Answer:
–12.6
Step-by-step explanation:
[tex] \frac{ \\ {x }^{3} - {2x}^{2} + kx - 12}{x - 5} = 0[/tex]
that means (x–5) is a factor of x³ – 2x² +kx – 12
since it resulted to zero according to the rules of zeros of polynomial.
x–5 = 0
x = 5
by substituting
f(x) = 5³ – 2(5)² + k(5) – 12 = 0
125 – 50 + 5k – 12 = 0
75 + 5k – 12 = 0
5k + 75 – 12 = 0
5k + 63 = 0
5k = –63
[tex] \frac{ \\ 5k}{5} = \frac{ - 63}{5} [/tex]
[tex]k = - 12.6 \: or \: - 12 \frac{3}{5} [/tex]
I hope I helped
What type of motion transforms the letter “p” into a “b”?
The type of motion which transforms the letter “p” into a “b” is; Reflection.
What type of motion transforms the letter “p” into a “b”?It follows that the task content demands that the type of motion which maps p to b be identified. On this note, it follows that the type of motion in discuss is Reflection. More particularly, a reflection over the horizontal axis most definitely causes this transformation.
Therefore, the motion which transforms the letter “p” into a “b” is; Reflection.
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A game has an expected value to you of $. It costs $ to play, but if you win, you receive $100,000 (including your $ bet) for a net gain of $. What is the probability of winning? Would you play this game? Discuss the factors that would influence your decision.
The probability of winning this game is 0.003
How to solve for the probability of winningLet us name p to be the probability of winning the game.
We have the expected value to be 100
100 = probability of winning + 1-p x loss
win = 100000 - 200
We then input in the formula
100 = p*99800 +(1-p) x -200
100 = 99800p - 200 + 200p
Collect like terms
100 + 200 = 99800 + 200p
300 = 100000p
Divide through by 100000
This give the
Probability of winning = 0.003
The probability of winning the game is very low hence I would not play.
The factors that would influence the decision are:
The cost of playingThe expected valueComplete questionA game has an expected value to you of ?$100. It costs ?$200 to? play, but if you win you receive? $100,000 (including your ?$200 ?bet), for a net gain of ?$99,800. What is the probability of? winning
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Aperson invest 10,000 shillings for 40 months compounded monthly. Find the accumulated amount If the rate is 5% per annum.
hi
monthly interest rate is : [tex]\sqrt[12]{1.05}[/tex]
So in forty month there will be : 10 000 * ( [tex]\sqrt[12]{1.05}[/tex] )^40 ≈11 776,06
On a piece of graph paper, plot the following points: A (3, 1), B (1, 5), C (9, 9), and D (11, 5). These coordinates will be the vertices of a quadrilateral. How would you use the distance formula and the slope formula to prove that this figure is actually a rectangle?
The given coordinates are actually a rectangle
How to determine the quadrilateral type?
The coordinates are given as:
A (3, 1), B (1, 5), C (9, 9), and D (11, 5).
Calculate the distance between the coordinates using:
[tex]d = \sqrt{(x_2 -x_1)^2 +(y_2 -y_1)^2[/tex]
So, we have:
[tex]AB = \sqrt{(3 -1)^2 +(1-5)^2} =\sqrt {20[/tex]
[tex]BC = \sqrt{(1 -9)^2 +(5-9)^2} =\sqrt {80[/tex]
[tex]CD = \sqrt{(9 -11)^2 +(9-5)^2} =\sqrt {20[/tex]
[tex]DA = \sqrt{(11 -3)^2 +(5-1)^2} =\sqrt {80[/tex]
The above shows that the opposite sides are congruent
Next, we calculate the slopes using:
m = (y2- y1)/(x2- x1)
So, we have:
AB = (1- 5)/(3-1) = -2
BC = (5- 9)/(1-9) = 1/2
CD = (9- 5)/(9-11) = -2
DA = (5- 1)/(11-3) = 1/2
The slopes of adjacent sides are opposite reciprocals.
This means that the sides are perpendicular
Hence, the given coordinates are actually a rectangle
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solve (6b^2+5)+(8b-20)
The sum of the given expression is 6b^2+8b-15
Sum of expressionGiven the following expression
(6b^2+5)+(8b-20)
We are to take the sum. On expansion;
6b^2+5 + 8b-20
Collect the like terms
6b^2+8b +5 - 20
6b^2+8b-15
Hence the sum of the given expression is 6b^2+8b-15
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How much will you have saved after 6 years by contributing $1,200 at the end of each year if you expect to earn 11% on the investment?
Savings after 6 years would be $ 1992.
What are savings?Savings is the money that remains after expenses and other commitments have been subtracted from income.
Savings are the sum of money that would otherwise be lying about, not being risked on investments or used for consumption.
Savings and investing can be compared because the latter includes putting money at risk in an effort to try to increase wealth.
Indicators of household debt or a negative net worth include negative savings.
According to the question,
Amount contributed(P)= $1200
Rate(R)= 11%
Time(T)= 6 years
Interest(SI)=P*R*T/100
= 1200*11*6/100
= 792
Savings after 6 years = P+ SI
= 1200 + 792
= $ 1992
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Continue preparing to analyze the data (step 3): Choose an appropriate numerical summary.
An appropriate numerical summary is descriptive statistics.
What is a numerical summary?It should be noted that a numerical summary means a number used to describe specific characteristics of a data set.
In this case, descriptive statistics can be used. This is used to describe the data set.
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The length of the major axis of the ellipse below is 14, and the length of the
red line segment is 3. How long is the blue line segment?
Answer:
11
Step-by-step explanation:
The focal length is 14, and thus the sum of the lengths of the blue and red segments is also 14.
In a one tail hypothesis where you only in the lower tail, what is the p-value if
P-Value becomes 0.8599 when (z < 1.08).
According to the statement
We have given that the Z-stat value is 1.08 and we have to find the P value with the help of Z-stat value.
So,
A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values.
and we know that
The P-Value is calculated by converting your statistic (such as mean / average) into a Z-Score.
So, If the z-stat is 1.08, the p-value for a left-tail test is the probability that (z < 1.08), which is 0.8599.
So, P-Value becomes 0.8599 when (z < 1.08).
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What is the area of a poster that is 1 1/2 feet by 2 1/2 feet?
The area of the poster is 28.75 feet square.
How to determine the areaNote that the area of a rectangle is given as;
Area = length × width
Length = 11. 5 feet
Width = 2. 5 feet
Area = 11. 5 × 2. 5
Area = 28. 75 feet square
Thus, the area of the poster is 28.75 feet square.
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What are the domain and range of g(x)=√x+4?
OD: [4, ∞) and R: [0, ∞)
OD: (-4, ∞) and R: (-∞, 0)
OD: [-4, ) and R: [0, ∞)
OD: (4,) and R: (-∞, 0)
Answer:
Domain [-4 , +∞)
Range [0 , +∞)
Step-by-step explanation:
the function g such that g(x)=√(x+4) ,is defined
for the values of x that verify :
x + 4 ≥ 0
⇔ x ≥ -4
Then
The domain of g is [-4 , +∞)
………………………………………………
Let x ∈ [-4 , +∞)
then x ≥ -4
then x + 4 ≥ 0
then g(x) = √(x+4) ≥ 0
Therefore, we can affirm that the range of g is [0 , +∞)
Solve the equation 5 (x + 1) = 5x
[tex]\textbf{Heya !}[/tex]
✏[tex]\bigstar\textsf{Given:-}[/tex]✏
an equation = [tex]\sf{5(x+1)=5x}[/tex]✏[tex]\bigstar\textsf{To\quad find:-}[/tex]
x = ?✏[tex]\bigstar\textsf{Solution \quad steps:-}[/tex] ✏
first use the distributive property
[tex]\sf{\longmapsto{ 5x+5=5x}[/tex]
subtract both sides by 5x
[tex]\sf{\longmapsto{0x+5=0}[/tex] (strange expression right)
[tex]\sf{\longmapsto 0=-5}[/tex] (whattt ?!)
the above statement's false, so the equation has no solutions
`hope that was helpful to u ~
;
Use the expression in the accompanying discussion of sample size to find the size of each sample if you want to estimate the
difference between proportions of men and women who own smartphones. Assume that you want 99% confidence that your
error is no more than 0.035.
Click the icon to view the discussion of sample size.
The sample should include men and
(Type whole numbers.)
women.
The sample should include 2702 men and 2702 women.
Given that you want to be 99% sure your error doesn't exceed 0.035.
A sample is a group of people selected from a larger population to provide data to researchers. The number of members of the population included in the sample is called the sample size.
The significance level is the probability that the test statistic is in the critical range if the null hypothesis is actually true. The confidence interval is 99% or 0.99.
The level of significance α = 1-0.99 = 0.01
the margin of error, ME = 0.035
The Z-critical value at the 99% confidence level or at the significance level of 0.01 is 2.575 (from standard normal tables).
We will now use the formula:[tex]n=\frac{Z^2_{critical}}{2(ME)^2}[/tex].
Substitute the values in the above formula, we get
n = (2.575) ² / (2 × (0.035) ²)
n = 6.630625 / (2 × 0.001225)
n = 6.630625 / 0.00245
n = 2701.966
Therefore, the sample should include 2702 males and 2702 females with a 99% certainty that the error does not exceed 0.035
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convert this into a standard number
[tex]10^{12}[/tex]
10^12 in standard number is 1000000000000
How to convert the number?The number is given as:
[tex]10^{12[/tex]
Expand the expression
[tex]10^{12} = 10 * 10 * 10 * 10 *10 * 10 * 10 * 10*10 * 10 * 10 * 10[/tex]
Evaluate the product
[tex]10^{12} = 1000000000000[/tex]
Hence, 10^12 in standard number is 1000000000000
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The diameter of a pipe is normally distributed, with a mean of 0.8 inch and a variance of 0.0004. What is the probability that the diameter of a randomly selected pipe will exceed 0.84 inch? (You may need to use the standard normal distribution table. Round your answer to three decimal places.) . Hint: take the square root of the variance to find the standard deviation.
The probability that the diameter of a randomly selected pipe will exceed 0.84 inch is 0.977
What is probability?Probabilities are used to determine the chances, likelihood, possibilities of an event or collection of events
How to determine the probability?The given parameters are:
Mean = 0.8
Variance = 0.0004
Calculate Standard deviation using
σ = √σ²
This gives
σ = √0.0004
Evaluate
σ = 0.02
Calculate the z-score at x = 0.84 using
z = (x - [tex]\bar x[/tex])/σ
This gives
z = (0.84 - 0.8)/0.02
Evaluate
z = 2
The probability is then represented as:
P(x > 0.84) = P(z > 2)
Next, we look up the value of the z table of probabilities
From the z table of probabilities, we have:
P(x > 0.84) = 0.977
Hence, the probability that the diameter of a randomly selected pipe will exceed 0.84 inch is 0.977
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Carlos spent 1\dfrac141
4
1
1, start fraction, 1, divided by, 4, end fraction hours doing his math homework. He spent 1\dfrac141
4
1
1, start fraction, 1, divided by, 4, end fraction as much time practicing piano.
How many hours did Carlos spend practicing piano?
write as a fraction
The time Carlos spend practicing his multiplication facts in fraction is 5/16 hours.
FractionA fraction is a value which has a numerator (upper or top value) divided by a denominator (lower or bottom value).
Time Carlos spent doing his math homework = 1 1/4 hoursTime spent practicing his multiplication facts = 1/4 of 1 1/4 hours
= 1/4 × 1 1/4
= 1/4 × 5/4
= (1 × 5) / (4 × 4)
= 5/16 hours
Therefore, the time Carlos spend practicing his multiplication facts is 5/16 hours.
Complete question:
Carlos spent 1 1/4 hours doing his math home work. He spent 1/4 of this time practicing his multiplication facts.
How many hours did Carlos spend practicing his multiplication facts?
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F(x) = | sin 3x | -3 find the maximum and minimum value
Answer:
-2 and -3
Step-by-step explanation:
The minimum of |sin 3x| is 0 and the max is 1 (since the sine function ranges from -1 to 1)
Therefore, the maximum is 1 - 3 = -2 and the minimum is 0 - 3 = -3.
which integer is closest to the third root of -23
Answer:
The number 23 is prime. Therefore, the cube root of 23 = ∛23 = 2.8439.
Step-by-step explanation:
What is
3/2^2 equal to
Answer: SQUARE ROOT OF 8
Step-by-step explanation: 2^3^1/2 is SQUARE ROOT OF 8. GOD BLESS
- FRIEND TO DINO
During an unusual storm, the temperature fell 8° C, rose 5° C, fell 4° C,
and then rose 6° C. If the temperature was 32° C at the outset of the storm,
what was it after the storm was over?
What are the solutions of the equation 9x4 - 2x2 - 7 = 0? Use u substitution to solve.
Intelligence test.
If you get it right ,you are a critical thinker.
You were in the garden, there are 34 people in the yard. You killed 30 .How many people are in the garden.
I
Answer: 5
Step-by-step explanation:
Because there are 35 when you add yourself so u subtract 30 to get 5
There would be 4 people left in the garden.
34 - 30 = 4.
Subtraction of NumbersSubtraction is a mathematical operation that involves taking away one number from another to find the difference between them. For the question, At first, there were 34 people in the garden and later 30 people were no longer alive in the garden, so the difference between 30 people and 34 people is 4.
Subtraction is the opposite of addition, and it is used to measure the distance between two values on the number line.
Subtraction is represented by minus Sign (-). For example, subtracting 5 from 10 gives you a difference of 5 (10 - 5 = 5).
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The amount of detergent dispensed into bottles of liquid laundry detergent bottles for a particular brand is normally distributed with a mean of 84.5 ounces with a standard deviation of 1.1 ounces. If seventeen bottles are randomly chosen from the factory, what is the probability that the mean fill is more than 84.8 ounces
The probability that the mean fill is more than 84.8 ounces is 0.39358
How to determine the probability that the mean fill is more than 84.8 ounces?From the question, the given parameters about the distribution are
Mean value of the set of data = 84.5Standard deviation value of the set of data = 1.1The actual data value = 84.8The z-score of the data value is calculated using the following formula
z = (x - mean value)/standard deviation
Substitute the given parameters in the above equation
z = (84.8 - 84.5)/1.1
Evaluate the difference of 84.8 and 84.5
z = 0.3/1.1
Evaluate the quotient of 0.3 and 1.1
z = 0.27
The probability that the mean fill is more than 84.8 ounces is then calculated as:
P(x > 84.8) = P(z > 0.27)
From the z table of probabilities, we have;
P(x > 84.8) = 0.39358
Hence, the probability that the mean fill is more than 84.8 ounces is 0.39358
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Find the area of the triangle
A
47°
b
32 ft
C
19 ft
Answer:
222 [ft²].
Step-by-step explanation:
1. the required area can be calculated according to the formula:
A=0.5*b*c*sin(A);
2. after substitution of b=32; c=19 and sin(A)≈0.7313537:
A=0.5*32*19*0.7313537=222.3315248 [ft²].
If sin(theta)=0.7 what is tan(theta)?
The value of tan theta from the given. expression is 7/√51
Trigonometry identityTrigonometry identity are expressions written in terms of sine, cosine and tangents.
Given the following trigonometry values as shown:
sin(theta)=0.7
Convert the decimal to fractions
sin(theta)= 7/10
According to SOH CAH TOA;
sin theta = opp/hyp
Determine the adjacent
adj = √hyp²-opp²
Substitute the values;
adj = √10²-7²
adj = √100-49
adj = √51
Determine the value of tan theta
tan(theta) = opp/adj
Substitute
tan(theta) = 7/√51
Hence the value of tan theta from the given. expression is 7/√51
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Fill in the missing term in the pair of equivalent ratios. *
5:30 15:
Your answer
The state low temperatures for five states are as follows: -42
Express v=
(2,1,5) as a linear combination of (1,2,1), (1,0,2) and (1,1,0)
The expression of v = (2,1,5) as a linear combination of (1,2,1), (1,0,2) and (1,1,0) is [tex](2,1,5)=1(1,2,1)+2(1,0,2)+-1(1,1,0)[/tex]
Linear CombinationsTo express v= (2,1,5) as a linear combination of (1,2,1), (1,0,2) and (1,1,0), it will be written as:
[tex](2,1,5)=x(1,2,1)+y(1,0,2)+z(1,1,0)[/tex]
This will yield the following equations:
x + y + z = 2........(1)
2x + z = 1..........(2)
x + 2y = 5........(3)
The solution to the system of equations above is:
x = 1, y = 2, z = -1
Therefore, the linear combination is:
[tex](2,1,5)=1(1,2,1)+2(1,0,2)+-1(1,1,0)[/tex]
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Money savers bank offers a savings account that earns 6% interest compounded continuously. If Manuel deposits $2200, how much will he have in the account after three years?
Answer:
2633.88
Step-by-step explanation:
Pe^rt
2200*e^.06*3
Resuelve los siguientes ejercicios:
Hallar la ecuación de la circunferencia:
a) Centro (2,-1) y radio 4
b) Centro C (1,3) y que pasa por el punto P (4, 6)
3. Hallar la ecuación de la parábola, las coordenadas de su foco y la longitud de su lado recto, si el vértice es el origen y pasa por el punto P:
a) Eje focal coincidente con el eje coordenado en x, P(2,4)
b) Eje focal coincidente con el eje coordenado en, P (6,3)
4. Hallar el foco, la ecuación de la directriz y la longitud del lado recto de las siguientes parábolas.
a) 4 x2 = 32y
b) 2y2 = -3x
5. Dadas las ecuaciones de las elipses, hallar las longitudes del semieje mayor y semieje menor, las coordenadas de los focos, los vértices y la longitud del lado recto.
a) 81 x2 + 144 y2 = 11664
b) 36x2+25y2 = 3600
Por restricciones de longitud no es posible resumir las respuestas asociadas a esta pregunta, invitamos cordialmente a leer la explicación para mayores detalles sobre el análisis de secciones cónicas.
¿Cómo analizar ecuaciones de secciones cónicas?
Según la geometría analítica, existen cinco tipos de secciones cónicas: (i) Circunferencia, (ii) Parábola, (iii) Elipse, (iv) Hipérbola, (v) Recta. 2) a) La ecuación estándar de la circunferencia se caracteriza con el centro (h, k) y la longitud del radio (r):
(x - h)² + (y - k)² = r²
(x - 2)² + (y + 1)² = 4²
b) La longitud del radio de la circunferencia se obtiene por el teorema de Pitágoras sobre la longitud del segmento CP:
r = √[(4 - 1)² + (6 - 3)²]
r = √(3² + 3²)
r = 3√2
(x - 1)² + (y - 3)² = 18
3) a) El eje focal forma parte del eje de simetría de la parábola. La ecuación estándar de la parábola es:
4 · p · x = y²
Donde p es la distancia entre el foco y el vértice.
Si tenemos que (x, y) = (2, 4), entonces la ecuación de la parábola es:
4 · p · 2 = 4²
p = 2
8 · x = y²
Las coordenadas del foco de la parábola son de la forma (x, y) = (h + p, k):
F(x, y) = (2, 0)
Ahora se determinan los extremos del lado recto: (x = 2)
8 · 2 = y²
y = ± 4
Los extremos del lado recto son (2, 4) y (2, - 4), cuya longitud de segmento es 8 unidades.
b) El eje focal forma parte del eje de simetría de la parábola. La ecuación estándar de la parábola es:
4 · p · x = y²
Si tenemos que (x, y) = (6, 3), entonces la ecuación de la parábola es:
4 · p · 6 = 3²
p = 8 / 3
(32 / 3) · x = y²
Las coordenadas del foco son F(x, y) = (8 / 3, 0).
Ahora se determinan los extremos del lado recto: (x = 6)
(32 / 3) · 6 = y²
y = ± 8
Los extremos del lado recto son (6, 8) y (- 6, - 8), cuya longitud de segmento es 16 unidades.
4) a) Tenemos una ecuación estándar de la forma 4 · p · y = x². A continuación, hallamos todas las variables requeridas:
x² = 8 · y
p = 2
Directriz: y = - 2, Foco: F(x, y) = (0, 2), Longitud del lado recto: 4
b) Tenemos una ecuación estándar de la forma 4 · p · x = y². A continuación, hallamos todas las variables requeridas:
y² = - (3 / 2) · x
p = - 3 / 8
Directriz: x = 3 / 2, Foco: F(x, y) = (- 3 / 2, 0), Longitud del lado recto: 3.
5) En esta parte debemos manipular algebraicamente las ecuaciones hasta su forma estándar para determinar los datos requeridos de cada caso. La ecuación estándar de la elipse tiene el siguiente problema:
(x - h)² / a² + (y - k)² / b² = 1
Donde:
(h, k) - Centro de la elipse.a, b - Longitudes de los semiejes.a) 81 · x² + 144 · y² = 11664
x² / 144 + y² / 81 = 1
x² / 12² + y² / 9² = 1
Longitud del semieje mayor: 12
Longitud del semieje menor: 9
c = √(12² - 9²)
c ≈ 7.937
Coordenadas de los focos: F₁ (x, y) = (- 7.937, 0), F₂ (x, y) = (7.937, 0)
Vértices: V₁ (x, y) = (- 12, 0), V₂ (x, y) = (12, 0)
Longitud del lado recto
144 · y² = 11664 - 81 · x²
y² = (11664 - 81 · x²) / 144
y = ± (1 / 12) · √(11664 - 81 · x²)
y = ± (1 / 12) · √(11664 - 81 · 7.937²)
y = ± 6.750
La longitud del lado recto es 13.5.
b) 36 · x² + 25 · y² = 3600
x² / 10² + y² / 12² = 1
Longitud del semieje mayor: 12
Longitud del semieje menor: 10
c = √(12² - 10²)
c ≈ 6.633
Coordenadas de los focos: F₁ (x, y) = (0, - 6.637), F₂ (x, y) = (0, 6.637)
Vértices: V₁ (x, y) = (0, - 12), V₂ (x, y) = (0, 12)
Longitud del lado recto
36 · x² = 3600 - 25 · y²
x² = 100 - (25 / 36) · y²
x = √[100 - (25 / 36) · y²]
x = √[100 - (25 / 36) · 6.637²]
x = ± 8.331
La longitud del lado recto es 16.662.
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Sarah had three times as many quarters as nickels. She had $3.20 in all. How many quarters did she have?
A. 12
B. 15
C. 4
Answer:
A. 12
Step-by-step explanation:
If she had 12 quarters, then she would have to have 4 nickels according to the problem.
12 * .25 = $3 (money in quarters)
4 * .05 = $.2 ( money in nickels)
Adding those 2 together, we get $3.2.
She have 12 quarters
what is unitary method?It is a method where we find the value of a single unit from the value of multiple units and the value of multiple units from the value of a single unit.
There are two situation in Unitary method:
There are 5 ice-creams. 5 ice-creams cost $125.
Step 1: Let’s find the cost of 1 ice cream. In order to do that, divide the total cost of ice-creams by the total number of ice-creams. The cost of 1 ice-cream = Total cost of ice-creams/Total number of ice-creams = 125/5 = 25. Therefore, the cost of 1 ice cream is $25.
Step 2: To find the cost of 3 ice-creams, multiply the cost of 1 ice cream by the number of ice-creams. The cost of 3 ice-creams is cost of 1 ice-cream × number of ice-creams = 25 × 3 = $75. Finally, we have the cost of 3 ice-creams i.e. $75.
Given:
If she had 12 quarters, then
12 * .25 = $3 ( quarters)
4 * .05 = $.2 ( in nickels)
Hence, she have 12 quarters.
Learn more about unitary method here:
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