The point of a square pyramid is cut off, making each lateral face of the pyramid a trapezoid with the dimensions shown. What is the area of one trapezoidal face of the figure?

Answer:

Option 2) [tex]a_n = -6(a_{n-1})[/tex]            

Step-by-step explanation:

We are given the following sequence in the question:

[tex]1, -6, 36, -216, ...[/tex]

We have to find the recursive relation for the sequence.

[tex]a_1 =1\\a_2 = -6 = -6(1) = -6(a_1)\\a_3 = 36 = -6(-6) = -6(a_2)\\a_4 = -216 = -6(36) = -6(a_3)[/tex]

Thus, continuing in the following manner, we get,

[tex]a_n = -6(a_{n-1})[/tex]

Thus, the recursive rule is given by

Option 2) [tex]a_n = -6(a_{n-1})[/tex]

Answer:

Step-by-step explanation:

The number of orange cans of paint is 6

Solution:

Given that,

[tex]\text{Number of cans of red paint } = 3\frac{1}{4}\\\\\text{Number of cans of yellow paint } = 3\frac{2}{12}[/tex]

Let us convert the mixed fractions to improper fractions

Multiply the whole number part by the fraction's denominator.

Add that to the numerator.

Then write the result on top of the denominator

[tex]\rightarrow 3\frac{1}{4} = \frac{4 \times 3 + 1}{4} = \frac{13}{4}\\\\\rightarrow 3\frac{2}{12} = \frac{12 \times 3 + 2}{12} = \frac{38}{12}[/tex]

Now we have to add red cans of paint and yellow cans of paint to get orange cans of paint

[tex]\text{Number of cans of orange paint } = \frac{13}{4} + \frac{38}{12}[/tex]

Take L.C.M for denominators

The prime factors of 4 = 2 x 2

The prime factors of 12 = 2 x 2 x 3

For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.

The new superset list is

2, 2, 3

Multiply these factors together to find the LCM.

LCM = 2 x 2 x 3 = 12

[tex]\text{Number of cans of orange paint } = \frac{13}{4} + \frac{38}{12}[/tex]

[tex]\text{Number of cans of orange paint } = \frac{13 \times 3}{4 \times 3} + \frac{38}{12}\\\\\text{Number of cans of orange paint } = \frac{39}{12} + \frac{38}{12}\\\\\text{Number of cans of orange paint } = \frac{77}{12} = 6.4 \approx 6[/tex]

Thus the number of orange cans of paint is 6

Answer:

in the year 2023

Step-by-step explanation:

Initial population of China = 1.355  billion

Initial population of India = 1.255  billion

Annual population growth rate of China = 0.44% = 0.0044

Annual population growth rate of India = 1.25% = 0.0125

Now,

Final population = P₀ [tex]\times e^{\text{rate}\times t}[/tex]

Here,

P₀ = initial population

t = time

Thus,

Population of India > Population of China

1.255[tex]\times e^{\text{0.0125}\times t}[/tex]  > 1.355[tex]\times e^{\text{0.0044}\times t}[/tex]  

or

[tex]e^{0.0081t}[/tex] > 1.07968

taking natural log both sides

0.0081t > ln (1.07968 )

or

0.0081t > 0.0766

or

t > 9.465

Hence,

9.465 year after 2014

i.e

in 2014 + 9.465 = 2023.46

in the year 2023

Using the formula for exponential growth and the provided population data for China and India, we can approximate that India's population will exceed China's in roughly 20 years.

In order to determine when India's population will exceed that of China, we can use the formula for exponential growth, which is P = P_0 * ert , where P is the future population, P_0 is the initial population, r is the growth rate, and t is time. As per the data provided, for China, P = 1.355 billion, r = 0.44%, and for India, P = 1.255 billion, r = 1.25%. We have to find time t when population of India will exceed that of China. This requires solving the equation: 1.355 * e0.0044t = 1.255 * e0.0125t.

This equation can be solved using algebra and logarithms. However, by making some approximations and using a spreadsheet or calculator, we can find that the population of India will exceed that of China after approximately 20 years, based on the growth rates provided.

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Answer:

D = A < D < B < C

Step-by-step explanation:

A = 4.6 x 10-4

Can be written as,

= 0.00046

B = 2.4 x 10-3

Can be written in this form,

= 0.0024

C = 3.5 x 105

Is written as,

350000

D = 6.3 x 10-4

Is also written as,

= 0.00063

A and D are in 4 decimal places and therefore from the 3th decimal place 46 is less than 63 so therefore 0.00046 is less than 0.00063.

B is greater than A and D because B which is 0.0024 is in 3 decimal places. C which is 35000 is the greatest because there are no decimal places and it is in tenth thousand.

So therefore,

A < D < B < C

Answer:

The fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots

Step-by-step explanation:

Given polynomial equation is [tex]X^4-4X^3-2X^2+12X+9=0[/tex]

And also given that 3,-1 and -1 are  the roots of the given polynomial equation

To find the fourth root of the polynomial equation and to solve the fourth root is real :

By synthetic division

_3|   1    -4    -2    12    9

       0    3    -3   -15    -9

___________________

_-1|       1     -1     -5    -3     0

            0     -1      2     3

___________________

            1      -2     -3     0

Therefore x-3 and x+1 is a factor

Therefore 3 and -1 are roots

Now we have the quadratic equation [tex]x^2-2x-3=0[/tex]

[tex](x+1)(x-3)=0[/tex]

Therefore x=-1,3 are the roots

Therefore the fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots.

Final answer:

The fourth root of the polynomial equation X⁴-4X³-2X²+12X+9=0 must be real because a polynomial of degree n has n roots, and since we already have real roots, the remaining root must also be real to have a pair. Upon analyzing, the fourth root is found to be 3.

Explanation:

The student has provided three roots of the fourth-degree polynomial equation X⁴-4X³-2X²+12X+9=0: 3, -1, and -1 (the latter being a repeated root). To determine why the fourth root must also be a real number, we can invoke the fundamental theorem of algebra, which states that a polynomial of degree n will have exactly n roots in the complex number system (including real and complex roots). Given that a polynomial with real coefficients will have complex roots that come in conjugate pairs, and since the known roots are all real, the unknown fourth root must also be real to satisfy the theorem.

Let's find the fourth root. The polynomial can be factored using the known roots:

Therefore, we have the equation (X-3)(X+1)²(X-a)=0, where 'a' is the unknown root. The product of the roots taken one at a time equals the constant term (9) of the polynomial with an alternate sign. This gives us the equation: 3 × -1 × -1 × a = 9. Solving for 'a' yields a=3, which is the fourth root.

Answer:

The weight of grapefruit is now 80 pound.

Step-by-step explanation:

Consider the provided information.

Let the x is the weight loss. The weight of grapefruit is 100 pounds and water is 92%. After evaporation water is 90%.

Thus the weight loss is:

[tex]0.92\times100-0.90(100 - x) = x[/tex]

[tex]92-90+0.90x=x[/tex]

[tex]2=x-0.90x[/tex]

[tex]2=0.1x[/tex]

[tex]x=20[/tex]

Hence, the weight loss is 80 pounds.

Therefore,  New weight is 100 - 20 = 80 pounds

The weight of grapefruit is now 80 pound.

Answer:

Step-by-step explanation:

Answers:

1) [tex]x^{8} y^{8}[/tex]

2) [tex]y^{3} \sqrt{y}[/tex]

3) [tex]5x^{4} \sqrt{6}[/tex]

4) [tex]\sqrt{7}[/tex]

5) [tex]\frac{\sqrt{z}}{z}[/tex]

Step-by-step explanation:

1) [tex]\sqrt{x^{16} y^{36}}[/tex]

Rewriting the expression:

[tex](x^{16} y^{36})^{\frac{1}{2}}[/tex]

Multiplying the exponents:

[tex]x^{\frac{16}{2}} y^{\frac{36}{2}}[/tex]

Simplifying:

[tex]x^{8} y^{8}[/tex]

2) [tex]\sqrt{y^{7}}[/tex]

Rewriting the expression:

[tex]\sqrt{y^{6} y}=(y^{6} y)^{\frac{1}{2}}[/tex]

Multiplying the exponents:

[tex]y^{\frac{6}{2}} y^{\frac{1}{2}}[/tex]

Simplifying:

[tex]y^{3} y^{\frac{1}{2}}=y^{3} \sqrt{y}[/tex]

3) [tex]\sqrt{150 x^{8}}[/tex]

Rewriting the expression:

[tex]\sqrt{(6)(25) x^{8}}[/tex]

Since [tex]\sqrt{25}=5[/tex]:

[tex]5x^{4}\sqrt{6}[/tex]

4) [tex]\frac{7}{\sqrt{7}}[/tex]

Multiplying numerator and denominator by [tex]\sqrt{7}[/tex]:

[tex]\frac{7}{\sqrt{7}} (\frac{\sqrt{7}}{\sqrt{7}})=\frac{7}{7\sqrt{7}}[/tex]

Simplifying:

[tex]\sqrt{7}[/tex]

5) [tex]\frac{5z}{\sqrt{25 z^{3}}}[/tex]

Rewriting the expression:

[tex]\frac{5z}{5z \sqrt{z}}[/tex]

Simplifying:

[tex]\frac{1}{\sqrt{z}}[/tex]

Since we do not want the square root in the denominator, we can multiply numerator and denominator by [tex]\sqrt{z}[/tex]:

[tex]\frac{1}{\sqrt{z}}(\frac{\sqrt{z}}{\sqrt{z}})[/tex]

Finally:

[tex]\frac{\sqrt{z}}{z}[/tex]

Using the process of Gaussian elimination, the system of linear equations is rewritten in the form of a matrix. It is then transformed into the Row-Echelon form, which helps determine possible solutions. The solution for this particular system of equations is x1 = 2, x2 = 2, and x3 = 1.

To solve this system of linear equations, you can use a process called

Gaussian elimination

. You start by rewriting the system in augmented matrix. Thus, the system

2x1 + x2 − 2x3 = 4

4x1 + 2x3 = 10

−4x1 + 5x2 − 17x3 = −15

becomes the matrix

[2 1 -2 4]

[4 0 2 10]

[-4 5 -17 -15]

The next step is to convert this matrix into the Row-Echelon form. Once you have a matrix in Row-Echelon form, you can easily see if there are any solutions by looking at the location of the zeros. If there is a row with all zeros on the left and non-zero terms on the right, then there is no solution. If there are infinite many solutions, its row will end with zeros. In this case, the solution is x1 = 2, x2 = 2, and x3 = 1.

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Question is not proper;Proper question is given below;

Noah has a total of 47 video games. he only buys action games and sports games. He has 21 more action games than sports games. how many action games, a, and sports games, s, does he own?

Answer:

Noah has 34 action games and 13 sports games.

Step-by-step explanation:

Given:

Total number of games he has = 47

Let the number of action games be 'a'.

Let the number of sports game be 's'.

So we can say that;

Total number of games he has is equal to sum of the number of action games and the number of sports games.

framing in equation form we get;

[tex]a+s = 47 \ \ \ \ \ eqaution \ 1[/tex]

Also Given:

He has 21 more action games than sports games.

so we can say that;

[tex]a=s+21 \ \ \ equation\ 2[/tex]

Now Substituting equation 2 in equation 1 we get;

[tex]a+s=47\\\\s+21+s=47\\\\2s+21=47[/tex]

Subtracting both side by 21 we get;

[tex]2s+21-21=47-21\\\\2s = 26[/tex]

Dividing both side by 2 we get;

[tex]\frac{2s}{2}=\frac{26}{2}\\\\s=13[/tex]

Now Substituting the value of 's' in equation 2 we get;

[tex]a=s+21=13+21=34[/tex]

Hence Noah has 34 action games and 13 sports games.

The value of [tex]\( x \) i[/tex]s irrelevant; the relationship between the areas remains constant regardless of its value.

To find the value of[tex]\(x\), let's denote the length of the ad as \(L\) and the width as \(W\). The area of the entire ad is \(L \times W\). Since the area of the photo is half the area of the ad, its area is \(\frac{1}{2} \times L \times W\).[/tex]

Now, we're given a diagram indicating that the length of the photo is [tex]\(x\) and its width is \(\frac{1}{2}W\). Therefore, the area of the photo is \(x \times \frac{1}{2}W\)[/tex].

We set up an equation based on the given information:

[tex]\[\frac{1}{2} \times L \times W = x \times \frac{1}{2}W\][/tex]

We cancel out the common factor of[tex]\(\frac{1}{2}W\) from both sides:\[L = x\][/tex]

This means that the length of the ad is equal to [tex]\(x\). Since we're not given any specific measurements or constraints on \(x\), its value could be any positive real number. Thus, the value of \(x\) is irrelevant to the relationship between the areas of the photo and the entire ad. Regardless of \(x\)[/tex], the area of the photo will always be half the area of the ad.

Final answer:

Increasing the sample size from 800 to 1,300 for estimating the mean weight of carry-on bags keeps the bias the same but decreases the variance, meaning that the sample will have lower variability around the true population mean.

Explanation:

When researchers working to estimate the mean weight of carry-on bags increase the sample size from 800 to 1,300, the correct effect on the bias and variance of the estimator is that the bias will remain the same and the variance will decrease. Bias is a measure of the systematic error of an estimator, and changing the sample size does not generally affect the estimator's systematic error if the estimator is unbiased to begin with. On the other hand, increasing the sample size leads to a decrease in variance which measures the spread of the sample means around the true population mean. Therefore, the larger the sample size, the closer the sampling distribution of the mean will be to the population mean, thus reducing variability, as indicated by a smaller standard deviation and a narrower confidence interval. Therefore, the correct answer is (C): The bias will remain the same and the variance will decrease.

y=2.50 units

Step-by-step explanation:

Given that angle ∠Y=75°, ∠Z=50°, side XY=2 units, and side XZ is y then applying the sine rule for this case,

x/sin ∠x =y/sin y =z/sin z

2/sin 50°=y/sin 75°

2 sin 75° =y sin 50°

y= 2 sin 75°/sin 50°

y=2.50 units

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Answer:

C

Step-by-step explanation:

I just finished the test :)

Answer:12 baskets must be sold in order to break even

Step-by-step explanation:

The materials needed by the fundraiser team to make each basket cost $3.75.

Let x represent the number of baskets that the team made and also sold. if they spent $75 to advertise their site, then the total cost for x baskets would be

3.75x + 75

The baskets are being sold for $10 each. It means that the total revenue would be

10x

In order to break even, the revenue must be equal to total cost. Therefore,

10x = 3.75x + 75

10x - 3.75x = 75

6.25x = 75

x = 75/6/25 = 12

The distance of a dog from a post and the cost of buying apples can be represented by functions. We can find the values of these functions by substituting specific values and calculating the corresponding outputs.

In this question, we are given a function that represents the distance of a dog from a post as a function of time. To find the value of f(20), we substitute 20 into the function and calculate the corresponding distance. To find the value of f(140), we do the same thing, substituting 140 into the function.

f(20) = 20 feet

f(140) = 140 feet

In the second part of the question, we are given a function C that represents the cost of buying n apples. To find the meaning of each expression or equation, we substitute the given value of n and calculate the corresponding cost.

C(8) = $4.50

C(2) represents the cost of buying 2 apples.

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Answer:

The answer to your question is  25 and 28

Step-by-step explanation:

Number 1 = x

number 2 = y

Conditions

1)                    x + y = 53         ------------- l

2)                   x - y =  3            ------------ ll

Solve this system of equations by elimination

                     x + y = 53

                     x - y =  3

                   2x      = 56

Solve for x

                      x = 56/2

                      x = 28

Substitute x in equation 2

                    28 - y = 3

                         - y = -28 + 3

                         - y = -25

                           y = 25

Answer: 21.9 pounds of his weight is made up of fat.

Step-by-step explanation:

The total weight of the individual is 129 pounds. The individual has a body fat percentage of 17.7%.

Therefore, the number of pounds of his body that is made up of fat would be

17.7/100 × 129 = 0.177 × 129 = 21.93 pounds.

Approximating to the nearest tenth, it becomes 21.9 pounds.

Answer:the first equation can be multiplied by 5 and the second equation by 3

Step-by-step explanation:

The operation that would create an equivalent system of equations with opposite like terms is that the first equation can be multiplied by 5 and the second equation by 3, the correct option is A.

The system of equation is set of equations which have a common solution.

The equations are

3x-3y = 3

4x+5y = 13

The value of x and y can be determined using Elimination Method.

In elimination method like terms have to be created to make an equivalent system,

The first equation can be multiplied by 5 and the second equation by 3 to solve the equations.

To know more about System of Equations

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Answer:

A

Step-by-step explanation:

The linear model can be assessed by the checking the independent variables having power 1 which shows the linear relationship between x and y. For example, as in the option B, C and D, the power of Xi's is one. Whereas the non linear model has the power for independent variables greater than 1. For example, as in option  A the model is a quadratic model because X associated with β2 has a power of a 2.

Thus the nonlinear model can be expressed as

Y = β0 + β1X + (β2X)2 + ε.

Answer:

4m width and 9m length

Step-by-step explanation:

Let the width of the rectangle be x

Length is 1 more than twice width= 1 + 2x

Area of rectangle is L * B

x(2x + 1) = 36

2x^2 + x = 36

2x^2 + x -36 = 0

2x^2 + 9x - 8x -36 = 0

Solving this:

(2x+9)(x - 4) = 0

X = 4 or -4.5

Distance cannot be negative, so x = 4m

The length is thus 2(4) + 1 = 9m

Final answer:

The width of the plot is 4 meters and the length is 9 meters.

Explanation:

To solve this problem, we can let the width of the plot be x meters. According to the problem, the length of the plot is one more than twice its width, so the length would be 2x + 1 meters. The area of a rectangle is given by the formula A = length * width. So we have the equation (2x + 1) * x = 36. Expanding and rearranging, we get 2x² + x - 36 = 0.

Factoring this quadratic equation, we get (2x + 9)(x - 4) = 0. Setting each factor equal to zero and solving for x, we find x = -4/2 and x = 4. Since the width cannot be negative, we discard x = -4/2 and conclude that the width of the plot is 4 meters. Substituting this value back into the equation for the length, we find the length is 2(4) + 1 = 9 meters.

Answer:the veterinarian made the first error in step 3

Step-by-step explanation:

the number of milligrams, m, varies directly with the weight of the dog, w.

Assuming constant of variation is k, then,

m = kw

k = m/w = 0.5/50 = 0.01

Therefore,

m = 0.01w

In step 3, Substituting 10 into the equation to find the dosage for a 10-pound dog like 10 = 0.01w was error.

The correct step is

m = 0.01 × 10

m = 0.1 milligrams

The distance between the base of the ladder and the wall is approximately 3.57 meters.

To find the distance between the base of the ladder and the wall, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the base of the ladder represents one side of the triangle, the height of the ladder represents another side, and the distance between the base of the ladder and the wall represents the hypotenuse.

Using the given information, we can calculate the distance as follows:

d^2 = 5^2 - 3.5^2

d^2 = 25 - 12.25

d^2 = 12.75

d ≈ √12.75

d ≈ 3.57m

Therefore, the distance between the base of the ladder and the wall is approximately 3.57 meters.

Answer:

a. is symmetric but not reflexive and transitive

b. is reflexive and transitive but not symmetric

c. is reflexive, symmetric and transitive

Step-by-step explanation:

The cases are missing in the question.

Let the cases be as follows:

a. R = {(1, 3), (3, 1), (2, 2)}

b. R = {(1, 1), (2, 2), (3, 3), (1, 2)}

c. R = ∅

R is defined on the set {1, 2, 3}

a.  R = {(1, 3), (3, 1), (2, 2)} is

b. R = {(1, 1), (2, 2), (3, 3), (1, 2)} is

is reflexive because (1, 1), (2, 2), (3, 3) is in R

is not symmetric because for (1,2) (2,1) is not in R

is transitive becaue for (1,1) and (1,2) we have (1,2) in R

c. R = ∅ is

reflexive, symmetric and transitive because it satisfies the definitions since there is no counter example.

Answer: None of the above

Step-by-step explanation:

Each of the presented points helps to describe how to collect and summarize data and how to make appropriate scientific inferences.

It provides a guide on how to use biostatistical principles with grounded mathematical and probability theory. It aims is to help understand and to interpret biostatistical analysis generally.

Answer:

8 weeks

Step-by-step explanation:

$125*8 = $1000

Answer:it will take you 8 weeks to have $1,000 in your savings account

Step-by-step explanation:

If you want to comeplete baby step 1 so that you have $1,000 in your savings account, and you are able to put in $125 a week. It means that the number of weeks that it will take you to have $1000 in your savings account would be

1000/125 = 8 weeks.

Subtracting a value from the input x shifts the graph that number of units to the right.

The answer would be C.

Answer:the difference between the cost of one evening ticket and one day time ticket s $1.8

Step-by-step explanation:

The cost of four evening movie tickets is $33.40. This means that the cost of one evening ticket would be

33.4/4 = $8.35

The cost of 6 daytime tickets is 39.30. This means that the cost of one daytime ticket would be

39.30/6 = $6.55

Therefore, the difference between the cost of one evening ticket and one day time ticket would be

8.35 - 6.55 = $1.8

Answer:

[tex]m = \dfrac{a^2-99}{2a}[/tex]

Step-by-step explanation:

on a number line, m is the point that is the midpoint of two other points.

the distance between each of the points to the midpoint is 10 units..

if a is the point 10 units less than m

and b is the point 10 units greater than m,

then,

[tex]m = a+10[/tex]

[tex]m = b-10[/tex]

we can add the two equations to form the midpoint formula.

[tex]2m = a+b[/tex]

we also know that the product of both numbers equal -99.

[tex]ab = -99[/tex]

we can substitute either 'a' or 'b' to the equation of m.

[tex]2m = a-\dfrac{99}{a}[/tex]

[tex]m = \dfrac{a^2-99}{2a}[/tex]

and this is the equation for the midpoint of the two numbers.

Answer:

c

Step-by-step explanation:

The minimum force required to open the jar using the wrench is 41.5 N, calculated based on the given torque of 8.9 N∙m and the effective radius of 0.2145 m.

Calculate the minimum force required to open the jar using the jar wrench:

1. Identify the torque required:

The problem states that the torque required to open the jar is 8.9 N∙m. This means that you need to apply a force that creates a twisting moment of 8.9 N∙m to overcome the friction between the lid and the jar.

2. Determine the effective radius:

The effective radius is the distance from the center of rotation (the center of the lid) to the point where the force is applied (the end of the wrench handle).

In this case, the effective radius is the sum of:

The length of the wrench handle (17 cm = 0.17 m)

Half the diameter of the lid (4.45 cm = 0.089 m / 2, assuming a circular lid)

So, the effective radius is 0.17 m + 0.0445 m = 0.2145 m.

3. Apply the torque formula:

The formula for torque is: τ = rF

τ = torque (in N∙m)

r = effective radius (in meters)

F = force (in Newtons)

You can rearrange this formula to solve for force: F = τ / r

4. Calculate the force:

Plug in the values: F = 8.9 N∙m / 0.2145 m

Calculate: F = 41.5 N

Therefore, the minimum force required to open the jar using the wrench is 41.5 N.

Answer:   2+i

================================

Work Shown:

-3 + 6i - (-5 - 3i) - 8i

-3 + 6i + 5 + 3i - 8i

(-3+5) + (6i+3i-8i)

2+1i

2+i

The simplified expression is in the form a+bi with a = 2, b = 1.

Answer:

2 + i.

Step-by-step explanation:

-3 + 6i - (-5 - 3i) - 8i       Distribute the negative over the parentheses:

= -3 + 6i + 5 + 3i - 8i

= - 3 + 5 + 6i + 3i - 8i     Now simplify like terms:

= 2 + i.