Step-by-step explanation:
We're asked to find the simplified form of,
[tex]\longrightarrow S=\underbrace{1^2+3^2+5^2+7^2+\,\dots}_{\textrm{$n$ terms}}[/tex]
The n'th term of the sequence [tex]1^2,\ 3^2,\ 5^2,\ 7^2,\,\dots[/tex] is [tex](2n-1)^2.[/tex] So,
[tex]\longrightarrow S=1^2+3^2+5^2+\,\dots\,+(2n-1)^2[/tex]
We know the following sum,
[tex]\longrightarrow 1^2+2^2+3^2+\,\dots\,+n^2=\dfrac{n(n+1)(2n+1)}{6}[/tex]
This is expression for the sum of squares of first 'n' natural numbers. In case of '2n' natural numbers,
[tex]\longrightarrow 1^2+2^2+3^2+\,\dots\,+(2n)^2=\dfrac{2n(2n+1)(4n+1)}{6}[/tex]
We separate odd squares and even squares in the LHS.
[tex]\begin{aligned}\longrightarrow\ \ &\Big(1^2+3^2+5^2+\,\dots\,+(2n-1)^2\Big)&\\+\ \ &\Big(2^2+4^2+6^2+\,\dots\,+(2n)^2\Big)&=\dfrac{2n(2n+1)(4n+1)}{6}\end{aligned}[/tex]
We can take 4 common from the sum of even squares.
[tex]\begin{aligned}\longrightarrow\ \ &\Big(1^2+3^2+5^2+\,\dots\,+(2n-1)^2\Big)&\\+\ \ &4\Big(1^2+2^2+3^2+\,\dots\,+n^2\Big)&=\dfrac{2n(2n+1)(4n+1)}{6}\end{aligned}[/tex]
Now the sums in LHS are, as written above,
[tex]1^2+3^2+5^2+\,\dots\,+(2n-1)^2=S[/tex]
[tex]1^2+2^2+3^2+\,\dots\,+n^2=\dfrac{n(n+1)(2n+1)}{6}[/tex]
So,
[tex]\longrightarrow S+\dfrac{4n(n+1)(2n+1)}{6}=\dfrac{2n(2n+1)(4n+1)}{6}[/tex]
[tex]\longrightarrow S=\dfrac{2n(2n+1)(4n+1)}{6}-\dfrac{4n(n+1)(2n+1)}{6}[/tex]
[tex]\longrightarrow S=\dfrac{2n+1}{6}\Big[2n(4n+1)-4n(n+1)\Big][/tex]
[tex]\longrightarrow S=\dfrac{2n+1}{6}\Big[4n^2-2n\Big][/tex]
[tex]\longrightarrow S=\dfrac{2n(2n-1)(2n+1)}{6}[/tex]
[tex]\longrightarrow\underline{\underline{S=\dfrac{n}{3}\,(4n^2-1)}}[/tex]
This is the expression for the sum of squares of first 'n' odd natural numbers.
A CD-Rom has 700 megabytes of storage space. What percent of the space is used by a file that takes up 154 megabytes?
A file that contains 154 megabytes occupies 22% of the available space.
How to calculate the percent of the space?We can calculate the percent of the space by using the following formula:
The percent of the space = used space / storage space × 100%
A CD-Rom holds 700 megabytes of data.
A file containing 154 megabytes occupies the space.
The percent of the space = used space / storage space × 100%
The percent of the space = 154 / 700 × 100%
The percent of the space = 0.22 × 100%
Apply the multiplication operation, and we get
The percent of the space = 22%
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15 circles are cut-out of a rectangle with a length of 10 inches and a height of 6 inches. the circles are side by side. for a class project, a teacher cuts out 15 congruent circles from a single sheet of paper that measures 6 inches by 10 inches. how much paper is wasted? (60 – 15π) square inches 15π square inches 45 square inches (60 – π) square inches
Answer:
[tex]60-15\pi[/tex] square inches
Step-by-step explanation:
The area of the paper is [tex](6)(10)=60[/tex] square inches.
The area of one circle is [tex]\pi(1^2)=\pi[/tex]. So, the combined area of the circles is [tex]15\pi[/tex].
Taking the difference yields [tex]60-15\pi[/tex].
Answer: its A
Step-by-step explanation:
edge 2023
please help me im struggling in math
Corresponding angles theorem states that parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
To prove m∠ = m∠2.
Then,
m∠3 + m∠2 = 180m∠1 + m∠3 = 180m∠1 = m∠2How to prove corresponding angles?The corresponding angles theorem states that If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Corresponding angles are formed when a transversal passes through two lines.
As we know that line segment m is parallel to the line segment n.
m || n .
Hence,
m∠3 + m∠2 = 180 (same interior angles)
Same side interior angles are two angles that are on the same side of the transversal and on the interior of (between) the two lines.
Same side interior angles are supplementary.
m∠1 + m∠3 = 180(sum of angles on a straight line)
Angles on a straight line is supplementary.
So, by transitive property of equality, we get:
m∠ = m∠2
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question 8 true/false: outliers should always be removed from the data set prior to final analysis. 1 point false; we only remove outliers if we have very good justification that suggests that removing the outlier is appropriate. true; outliers distort model fit and must be removed to assure reliable results. false; we only remove outliers after checking to make sure doing so drastically improves model fit.
False; we only remove outliers if we have a very good justification that suggests that removing the outlier is appropriate.
You should discard the outlier if it is clear that it resulted from inaccurately measured or recorded data. You may discard the outlier if it has no effect on the results but has an impact on the assumptions. Outliers might occasionally point to a data-gathering error. However, there are situations when they might have an impact on data collection, making them crucial to keep in mind in order to fully comprehend the information. Eliminating outliers gives some of my variables a normal distribution and improves the effectiveness of modifications for the other variables. As a result, it would appear that eliminating outliers before transformation is the best course of action.
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You have 1 full pizza, out of which your friend ate one hundredth ix time from it. There i ome part of pizza left for you. How much you will get to eat ?
Pizza left for me equal to 0.94 part
Both rates and ratios are a comparison of two numbers.
A rate is simply a specific type of ratio.
The difference is that a rate is a comparison of two numbers with different units, whereas a ratio compares two numbers with the same unit.
For example, in a room full of students, there are 10 boys and 5 girls. This means the ratio of boys to girls is 10:5.
If we simplify the ratio, we see that the ratio of boys to girls is 2:1, since
10÷5=2and 5÷5=1
. Thus, there are 2 boys in the room for every 1 girl.
A friend ate 1/100 part of pizza 6 times
Totally , He have eaten= 1/100 * 6
=3/50
3/50 part of pizza he eaten = 0.06
Total pizza = pizza eaten + pizza left
pizza left = Total pizza - pizza eaten
= 1 - 0.06
= 0.94
pizza left for me equal to 0.94 part
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For example, in a room full of students, there are 10 boys and 5 girls. This means the ratio of boys to girls is 10:5.
If we simplify the ratio, we see that the ratio of boys to girls is 2:1, since
10÷5=2and 5÷5=1
Thus, there are 2 boys in the room for every 1 girl.
A friend ate 1/100 part of pizza 6 times
Totally , He have eaten= 1/100 * 6
=3/50
3/50 part of pizza he eaten = 0.06
Total pizza = pizza eaten + pizza left
pizza left = Total pizza - pizza eaten
= 1 - 0.06
= 0.94
pizza left for me equal to 0.94 part
(3x-5) + (5 + 6x) need answer fast
Answer:
Step-by-step explanation:Rearrange terms
3x-5+({5+6x})
Solution
9x will that help?
Answer:
Step-by-step explanation:
(3x-5)+(5+6x)
3x-5+5+6x
3x+6x+5-5
3x+6x
9x
i think this is the right answer
How do you graph a line with two independent variables?
To graph a line with two independent variables, you can use a three-dimensional coordinate system. The three dimensions of the coordinate system are the two independent variables and the dependent variable.
For example, suppose you have a line described by the equation y = 2x + 3z + 4. In this equation, x and z are the independent variables and y is the dependent variable. To graph this line, you would need to plot points in three-dimensional space, using the values of x, y, and z from the equation.
To do this, you can choose specific values for x and z, substitute those values into the equation to find the corresponding value of y, and plot those three values as a point in three-dimensional space. You can repeat this process for several different values of x and z to plot multiple points on the line. Once you have plotted enough points, you can connect them with a line to graph the equation in three dimensions.
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Find the volume of the solid below. Round your answer to the nearest tenth if necessary.
Width 13 m
Length 17 m
Hight 2m
What’s the volume
Answer: 442
Step-by-step explanation: volume is length x width x height
so plug it in to get 17 x 13 x 2 = 442
in a football game all of the home team points are from 7 point touchdowns and 3 point field goals. the team scores six times write and solve a system of linear equations fo find th enumbers of touchdowns and field goals that the hom team scores
The system of Linear equations is; y = 7x + 3(6 - x) and the solution is;
7 scores from touchdowns
15 scores from field goals
Total score of 22
How to solve a system of linear equations?Let the number of touchdowns be x
Let the total of the home team points be y
Thus, the number of field goals = 6 - x since the team scores 6 times.
Thus, since the home team points are from 7 point touchdowns and 3 point field goals, then it means we have the equation;
y = 7x + 3(6 - x)
where;
7x is scores from touchdowns
3(6 - x) is scores from field goals
Thus;
y = 7x + 18 - 3x
y = 4x + 18
Let's try x = 1 and we have;
7(x) = 7(1) = 7 scores from touchdowns
3(6 - 1) = 15 scores from field goals
y = 4(1) + 18
y = 22
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What is the measure of the central size of this rotation?
The value of the angle of the arc in the wheel is equal to 135. The correct option is D.
What is an angle of an arc?An arc of a circle is a section of the circle's circumference between two radii. A circle's central angle is an angle formed by two radii with the vertex in the centre. The central angle of an arc is the central angle that the arc subtends. The length of an arc is equal to the length of its central angle.
The angle suspended by the arc on the wheel is 3/8 that of the one revolution of the wheel.
The value of the angle will be calculated as,
Angle = 3/8 x 360
Angle = 3 x 45
Angle = 135
Therefore, the value of the angle of the arc in the wheel is equal to 135. The correct option is D.
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Which table represents a linear function?
Answer:
f(x)
Step-by-step explanation:
f(x) changes at a constant rate when x changes from 1 to 2 the change in y = 7 when x changes from 2 to 3 the change in y = 7 etc
the others do not change at a constant rate when x changes by one
The data represented by table number one is showing the linear relation.
What is linear relation?A linear relationship (or linear association) is a statistical term that describes a two-variable relationship that follows a straight line.
The momentum of a variable is represented by the rate of change, which is used to mathematically describe the percentage change in value over a defined period of time. If the rate of change is constant then the relationship will be linear.
For table number one the data is,
x F(x)
1 12
2 19
3 26
4 33
The rate of change will be,
R₁ = ( 19 - 12 ) / ( 2 - 1 ) = 7
R₂ = ( 26 - 19 ) / ( 3 - 1 ) = 7
R₃ = ( 33 - 26 ) / ( 4 - 3 )= 7
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What is the relationship between the volume of a prism and pyramid with congruent bases and heights?
The relationship between the volume of a prism and pyramid with congruent bases and heights is that the volume of pyramid is one-third of the volume of prism.
As the prism and pyramid have congruent bases and heights, let us consider the area of the triangular base is A and the height is h.
For a prism of base area A and height h, volume is determined by the formula
Vpm = A×h
For a pyramid of base area A and height h, volume is determined by the formula
Vpd = A×h/3
From both relations for volume
Vpd = Vpm/3
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help with this problem please
How did slavery benefit the industrial revolution?
The provision of a necessary raw material for industrial production by industrial revolution benefitted for slavery.
what is industrial revolution?Agrarian and handicraft economies were replaced by industrial and machine production during the Industrial Revolution in modern history. These technical developments brought about new methods of working and living that dramatically altered civilization.
the provision of a necessary raw material for industrial production by industrial revolution benefitted for slavery.
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4x+3y=23 x-5y=0 substitution
The required solution of given equation is x = 5, y = 1.
What is substitution method to equation?The effective use of substitution relies upon two things: first, given a circumstance in which factors happen, a replacement is just a difference in factor; second, it is just successful if the difference in factor improves on the circumstance and, ideally, empowers one to take care of the worked on issue.
According to question:We have,
4x+3y=23---------()1
And
x-5y=0 ⇒ x = 5y
Put the value of x in equation(1)
4(5y) + 3y = 23
20y + 3y = 23
23y = 23
y = 1
Then x = 5y = 5(1) = 5
Thus, solution of equation x = 5, y = 1
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√2x+6=-8
Please help!!
Answer:
no solution
Step-by-step explanation:
You want the solution to the radical equation √(2x+6) = -8.
AnalysisThe radical symbol is used to signify the positive square root. Whatever non-negative value (2x+6) may have, the left side expression returns a positive value for the root. It can never be equal to -8.
There is no solution to the given equation.
sales at a fast-food restaurant average $7,800 per day. the restaurant decided to introduce an advertising campaign to increase daily sales. to determine the effectiveness of the advertising campaign, a sample of 64 days of sales were taken. they found that the average daily sales were $8,080 per day. from past history, the restaurant knew that its population standard deviation is about $1,000. the value of the test statistic is .
Using sample standard deviation formula and the provided sample mean value, The test statistic are $8,080(sample mean) and 125 (sample standard deviation).
What do you mean by statistics?Large-scale numerical data collection and analysis, especially with the aim of extrapolating proportions in the total from those in a representative sample.
What do you mean by population and sample?A population is the entire set of people in a group, whether that group is a country or a collection of people who share a certain trait.
A sample is a condensed, controllable representation of a larger group. It is a subgroup of people with traits from a wider population. When population sizes are too big for the test to include all potential participants or observations, samples are utilized in statistical testing.
Sample mean= $8,080
population standard deviation= $1,000
number of sample=64
sample standard deviation = [tex]\frac {1000}{\sqrt {64}}[/tex]
=1000/8
=125
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A driving service, like Uber or Lyft, charges the following, as modeled in the table, per mile driven.
Miles Driven Cost
0 $10
1 10.70
2 11.40
3 12.10
4 12.80
What is the rate of change/cost per mile driven?
What does the first point (0, 10) represent in real life terms?
Using m to represent miles and C(m) to represent the cost, what equation models the function in the table?
Answer:
The rate of change/cost per mile driven can be calculated by finding the slope of the line that connects the points (1,10.70) and (0,10). The slope is the change in the y-value divided by the change in the x-value, so the slope of the line is (10.70-10)/(1-0) = $0.70/mile.
The point (0, 10) represents the cost of the driving service when 0 miles are driven. In other words, the service costs $10 for no miles driven.
The equation that models the function in the table is C(m) = $10 + $0.70m. The first term represents the cost of the service when no miles are driven and the second term represents the cost per mile driven.
What score has a mean of 50 and a standard deviation of 10?
T-scores show how many standard deviation units an examinee's score differs from the mean. Because T-Scores have a mean of 50 and a standard deviation of 10, any T-Score may be directly interpreted.
What is T-score?A t-score (sometimes known as a t-value) is the number of standard deviations from the mean of the t-distribution. The test statistic used in t-tests and regression testing is the t-score. It may also be used to represent how distant an observation is from the mean when the data follows a t-distribution. A t test is a statistical test used to compare two groups' means. It is frequently used in hypothesis testing to assess whether a procedure or treatment has an effect on the population of interest, or whether two groups vary.
Here,
T-scores reflect how many standard deviation units an examinee's score is above or below the mean. Because T-Scores always have a mean of 50 and a standard deviation of 10, any T-Score may be directly interpreted.
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How do you find the solution to a system of infinite solutions?
To solve systems of an infinite equation we need to determine whether the equation is dependent or independent and consistent, or inconsistent.
The number of solutions in an equation depends on the total number of variables contained in it is known as Infinite solutions.
The system of equations contains two or more equations that have two or more variables. It can be any combination such as
2 equations in 3 variables.6 equations in 4 variables.If a pair of linear equations have unique or infinite solutions, then the system of equations is said to be a consistent pair of linear equations.
For example, consider two equations in two variables as follows:
a1x + b1y = c1 ——- (1)
a2x + b2y = c2 ——- (2)
The above-given equations are consistent and dependent and have infinitely many solutions.
(a1/a2) = (b1/b2) = (c1/c2)
There are a few conditions for infinite solutions:
The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept.If the two lines have the same y-intercept and slope, they are actually in the same exact line.To know more about infinite solutions:
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Ice cream is packaged in cylindrical gallon tubs. A tub of ice cream has a total surface area of 387.79 square inches.
If the diameter of the tub is 10 inches, what is its height? Use π = 3.14.
The required height of the tub is given as 7.32 inches.
As mentioned in the question, Ice cream is packaged in cylindrical gallon tubs. A tub of ice cream has a total surface area of 387.79 square inches, and the diameter of the tub is 10 inches.
What is surface area?The surface area of any shape is the area of the shape that is faced or the Surface area is the amount of area covering the exterior of a 3D shape.
Here,
The surface area of the cylinder = 2πrh + 2πr²
Substitute the values,
387.79 = 2 × 3.14 × 5 [h + 5]
h = 7.32 inches
Thus, the height of the cylindrical gallon tub is given as 7.32 inches.
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Answer:
7.35 inches
Step-by-step explanation:
I took the test and got it right :)
Write the standard form of the equation of a parabola passing through the point (1, 32) with a minimum point at (-2, 5).
Answer:
[tex]y=3x^2+12x+17[/tex]
Step-by-step explanation:
The equation in vertex form is [tex]y=a(x+2)^2+5[/tex].
We can use the point [tex](1,32)[/tex] to solve for [tex]a[/tex].
[tex]32=a(1+2)^2+5 \\ \\ 27=9a \\ \\ a=3 \\ \\ \\ \therefore y=3(x+2)^2 + 5 \\ \\ =3(x^2+4x+4)+5 \\ \\ =3x^2+12x+12+5 \\ \\ =3x^2+12x+17[/tex]
What is the slope of the equation 4x 2y =- 5?
the slope is 5/-2 so this line "cuts" the y axis at y=-2.50000
y-intercept = 5/-2 = -2.5
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
-2*y-(-4*x+5)=0
Pulling out like terms
Pull out like factors :
-2y + 4x - 5 = -1 • (2y - 4x + 5)
Equation at the end
Equation of a Straight Line
Solve -2y+4x-5 = 0
"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.
In this formula :
y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis
The X and Y intercepts and the Slope are called the line properties. We shall now graph the line -2y+4x-5 = 0 and calculate its properties
Graph of a Straight Line :
Calculate the Y-Intercept :
Notice that when x = 0 the value of y is 5/-2 so this line "cuts" the y axis at y=-2.50000
y-intercept = 5/-2 = -2.5
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Answer question please
Answer:
Step-by-step explanation:
c=84
c+44 is going to be equal to the sum of c and c-40 because c+44 is the remote angle to both c-40 and c so you set it up as:
c + c - 40= c + 44
you simplify
2c - 40 = c + 44
you add 40 to both sides
2c = c + 84
you subtract c
and your answer is: c=84
How do you prove AAS congruence rule?
The AAS congruence rule states that if two angles and the included side of a triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent.
What is AAS congruence rule?The AAS congruence rule is a rule used in geometry to determine when two angles are congruent, or equal. This rule states that two angles are congruent if they are both angles of a triangle and they share the same side or vertex. This rule is a part of the Angle-Angle-Side (AAS) theorem, which states that three angles and three sides of one triangle are equal to three angles and three sides of another triangle.
To prove this congruence rule, we need to use the Side-Angle-Side (SAS) congruence theorem. This theorem states that if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent. This theorem can be used to prove the AAS congruence rule, as it states that if two angles and the included side of a triangle are congruent to the corresponding parts of another triangle, then they can be used to form two sides and the included angle of a new triangle, which satisfies the conditions of the SAS congruence theorem. Therefore, the two triangles are congruent, proving the AAS congruence rule.
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Mikaela is completing in a race in which she both runs and rides a bicycle. She runs 5 kilometers in 0.5 hour and rides her bicycle 20 kilometers in 0. 8 hour.
At the rate given, how many kilometers can Mikaela run in 1hour?
At the rate given, how many kilometers can Mikaela bike in 1 hour?
If Mikaela runs for 1 hour and bikes for 1 hour at the rates given, how far will she travel
Answer:
Mikaela run 10 km in 1hour
Mikaela bikes 25km in 1hour
35km If Mikaela runs for 1 hour and bikes for 1 hour at the rates given
Step-by-step explanation:
Let's solve how many km Mikaela runs in 1 hour.
0.5 hour = 30 minutes
5km = 30 minutes
10km = 60 minute
Let's solve how many km Mikaela bikes in 1 hour.
0.8 hour = 48 minutes
20km = 48 minutes
To get from 48 minutes to 60 minutes, we times 1.25. So, we take 20 times 1.25 = 25km
So, Mikaela bikes 25km in 1 hour.
Let's solve how far she traveled If Mikaela runs for 1 hour and bikes for 1 hour at the rates given
10km+ 25km = 35km
Pregunta 2
La recta L3 pasa por la intersección de las rectas L₁ : 2x + 3y - 8 = 0 y L2: 2x 9y+4= 0 y por el
punto (1; 2). Determinar la ecuación de L3.
-
The linear function for Line L3 is given as follows:
y = -2/3x + 8/3.
How to define the linear function?The lines L1 and L2 are given as follows:
L1: y = -2/3x + 8/3.L2: y = -2/9x - 4/9.The x-coordinate of intersection is obtained as follows:
-2/3x + 8/3 = -2/9x - 4/9.
Multiplying everything by 9, the expression is given as follows:
-6x + 24 = -2x - 4.
4x = 28
x = 7.
The y-coordinate is given as follows:
y = -2/3(7) + 8/3 = -2.
Then the points on line L3 are given as follows:
(7,-2) and (1,2).
Given two points, the slope is calculated by the division of the change in y by the change in x, hence:
m = (-2 - 2)/(7 - 1)
m = -2/3.
Then:
y = -2/3x + b
When x = 1, y = 2, then the intercept b is calculated as follows:
2 = -2/3 + b
b = 2 + 2/3
b = 8/3.
Thus:
y = -2/3x + 8/3.
What is the translation?The problem asks for the definition of the line L3, which passes through the intersection of the linear L1 and L2 and through point (1,2).
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kate was trying to decide which type of frozen pizza to restock based on popularity: pepperoni pizza or sausage pizza. after studying the data, she noticed that pepperoni flavors sold best on the weekdays and on the weekends, but not best overall. which paradox has kate encountered? false negative benford's law false positive simpson's paradox save
Kate has encountered Simpson's Paradox because even though pepperoni pizza is the most popular flavor on weekdays.
Simpson's Paradox is the phenomenon that occurs when a trend appears to be in one direction when data is grouped in one way, but the trend is reversed when the data is grouped in another way.
In this case, Kate has encountered Simpson's Paradox because even though pepperoni pizza is the most popular flavor on weekdays, and again on the weekends when all the data is considered, the overall popularity of sausage pizza is greater.
Simpson's Paradox can be explained by the fact that the number of sausage pizza sales may be greater than the number of pepperoni pizza sales, even if the percentage of pepperoni pizza sales is higher.
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Kate has encountered the Simpson's Paradox. By definition, Simpson's Paradox is a phenomenon in which a trend appears in several different groups of data but disappears when these groups are combined. Option c is the right option
A relationship between two variables in a population might exist, vanish, or even change when the population is split up into subpopulations, a statistical phenomenon is known as Simpson's paradox.
Simpson's paradox is a statistical effect that arises when the marginal association between two categorical variables differs qualitatively from the partial association between the same two variables after adjusting for one or more other factors.
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what expression is equal to x^5 • x^2
Which of the following lists all the roots of x^4 =x^3 +x^2 + x+2?
F {−1, 2}
H {−2, −1}
G {−1, 2, i, −i}
J {−2, −1, i, −i}
Answer:
G
Step-by-step explanation:
[tex]x^4-x^3-x^2-x-2=0 \\ \\ (x+1)(x^3+x-2x^2-2)=0 \\ \\ (x+1)(x(x^2+1)-2(x^2+1))=0 \\ \\ (x+1)(x-2)(x^2+1)=0 \\ \\ (x+1)(x-2)(x-i)(x+i)=0 \\ \\ x=-1, 2, i, -i[/tex]