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What is the value of A when we rewrite 3^x as A^5x

The roots using the trigonometric formula is -2 + √3

The cubic formula is ax3 + bx2 + cx + d = 0. There is a wondering relation between the roots and the coefficients of a cubic polynomial.

The given function is

f(x) = x3 – 15x – 4

Using the Cardanos method we have

[tex]\sqrt[3]{2+11i} + \sqrt[3]{2-11i}[/tex]

Recall that the sum of the cubic root u of 2+11i with a cubic root u of 2-11i

Such that uv = -15/3 = 5

Now take u = 2+i and v = 2-i The indeed u³ = 2+11i, v³ = 2+11i and uv = 5

Therefore, 4(-u+v) is a root

But now take ω = -1/2 + √3/2i, Then ω² = -1/2 - √3i/2, ω = 1

and if you take u' = ωu, v' ω²v

u'' = ω²u, and v'' = ∈v

Then u' +v and u'' +v'' will be roots too

This means that -2±√3, v' + u' = -2 √3 and u'' + v'' = -2 +√3

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The assembly time for a product is uniformly distributed between 6 to 10 minutes.

The probability of assembling the product in 8 minutes or less is 0.5 (option c).

Solution: Given, the assembly time for a product is uniformly distributed between 6 to 10 minutes. The range is a = 6 to b = 10.The probability of assembling the product in 8 minutes or less is to be determined.

Let's calculate the probability using the formula:  P(x < or = 8) = (x - a) / (b - a)Here, a = 6, b = 10, and x = 8.P(x < or = 8) = (8 - 6) / (10 - 6) = 2 / 4 = 0.5Therefore, the probability of assembling the product in 8 minutes or less is 0.5. So, the correct option is (c) 0.5.

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To evaluate the integral ∫ x ds, we need to parameterize the given curves and compute the arc length integral. In part (a), we evaluate the integral for the straight line segment from (0, 0) to (12, 6). In part (b), we evaluate the integral for the parabolic curve from (0, 0) to (2, 12).


(a) For the straight line segment x = t, y = t/2 from (0, 0) to (12, 6), we can parameterize the curve as follows: x = t, y = t/2. The differential arc length element ds is given by ds = √(dx² + dy²). Substituting the parameterizations, we have ds = √(dt² + (dt/2)²) = √(5/4 dt²). Thus, the integral becomes ∫ x ds = ∫ t √(5/4 dt²) = ∫ t (√5/2) dt. Integrating with respect to t from 0 to 12, we get (√5/2) ∫ t dt = (√5/2) (t²/2) evaluated from 0 to 12. Evaluating this expression, we find that the integral is equal to (√5/2) (144/2) = 36√5.
(b) For the parabolic curve x = t, y = 3t² from (0, 0) to (2, 12), we can parameterize the curve as before: x = t, y = 3t². The differential arc length element ds is given by ds = √(dx² + dy²). Substituting the parameterizations, we have ds = √(dt² + (6t dt)²) = √(1 + 36t²) dt. Thus, the integral becomes ∫ x ds = ∫ t √(1 + 36t²)dt. Integrating with respect to t from 0 to 2, we can use techniques like substitution or numerical methods to evaluate the integral and obtain the result.

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a) The height of the ball after 2 seconds is 7.6 metres.

b) The given function is f(x) = (x - 2).

a) The inverse function of f(x) is g(x) = x + 2.

b) A farmer has 600 metres of fencing to enclose a rectangular area and divide it into 14 sections.

a) The total area enclosed can be expressed as a function of the length of the rectangle x as A(x) = 300x - x².

b) The domain of the area function is (0, 300) and its range is [0, ∞).

a) h(t) = -4.9t² + 13.1t + 1 When the time, t = 2 seconds;

the height of the ball, h(t) =h(2) = -4.9(2)² + 13.1(2) + 1= -4.9(4) + 26.2 + 1= -19.6 + 27.2= 7.6 metres

Therefore, the height of the ball after 2 seconds is 7.6 metres.

b) h(t) = -4.9t² + 13.1t + 1When the height of the ball, h(t) = 0.5 metres; we can write the equation as:

0.5 = -4.9t² + 13.1t + 1

Rearranging,4.9t² - 13.1t + 0.5 - 1 = 0i.e. 4.9t² - 13.1t - 0.5 = 0

Solving using the quadratic formula, t = [-(-13.1) ± √(13.1² - 4(4.9)(-0.5))]/(2(4.9))= [13.1 ± √(171.61)]/9.8≈ 0.138 and t ≈ 2.23

Thus, the ball reaches a height of 0.5 metres twice, at approximately 0.14 seconds and 2.23 seconds.

The given function is f(x) = (x - 2).

a) We can find the inverse of a function by interchanging x and y and then solving for y.

Here, f(x) = (x - 2) so we can write it as x = (y - 2)

Interchanging x and y: x = (y - 2)

Solving for y, we get: y = x + 2

Hence, the inverse function of f(x) is g(x) = x + 2.

b) Domain and range of the function and its inverse

The domain of f(x) is all real numbers. (i.e. -∞ < x < ∞)

The range of f(x) is all real numbers. (i.e. -∞ < f(x) < ∞)

The domain of g(x) is all real numbers. (i.e. -∞ < x < ∞)

The range of g(x) is all real numbers. (i.e. -∞ < g(x) < ∞)

The steps of the solution are given below:

Therefore, the inverse function of f(x) is g(x) = x + 2.

The domain of f(x) is all real numbers, and its range is all real numbers.

Similarly, the domain and range of g(x) is all real numbers.

A farmer has 600 metres of fencing to enclose a rectangular area and divide it into 14 sections.

Let x be the length and y be the width of the rectangle.

a) Write an equation to express the total area enclosed as a function of the length x.

Perimeter of the rectangle = Total fencing = 600 metres2(x + y) = 600i.e. x + y = 300

Solving for y, we get:

y = 300 - x

Thus, the area of the rectangle is A(x) = xy = x(300 - x) = 300x - x²

Therefore, the total area enclosed can be expressed as a function of the length of the rectangle x as A(x) = 300x - x².

b) Determine the domain and range of the area function.

The domain of the function is the set of all possible values of x, which in this case is the set of all positive real numbers. (i.e. 0 < x < 300)

The range of the function is the set of all possible values of A(x), which in this case is the set of all non-negative real numbers. (i.e. 0 ≤ A(x) < ∞)

Therefore, the domain of the area function is (0, 300) and its range is [0, ∞).

The steps of the solution are given below:

Therefore, the total area enclosed can be expressed as a function of the length of the rectangle x as A(x) = 300x - x². The domain of the area function is (0, 300) and its range is [0, ∞).

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Answer:

13.93

Step-by-step explanation:

see attached for explanation

Answer:

B. 7 people

Step-by-step explanation:

If you multiply 8x16 you get: 128. Then you subtract 135 from 128 and get: 7.  Therefore, 7 people will not be able to go if they only rent 16 minivans.

Answer:

2y + 10 / 5

Step-by-step explanation:

Quotient tells you you're dividing. Sum of 10 means add that to whatever else they say. Twice y = 2y.

Please lmk if you have questions.

Answer:

Step-by-step explanation:

455

Answer:

Answer is Option A

Step-by-step explanation:

the things people do for points smh :/

The rectangular equation which represents the parametric equations; x = t^(¹/2) and y = 4t is; y = 4x2, for x ≥ 0.

From the task content, it follows that the parametric equations given are;

Hence, it follows that; t = x² and y= 4t

Ultimately, upon substitution of x² for t; the resulting rectangular equation is; y = 4t².

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The given statement, "predictive modelling and lifetime value modelling are the same" is false. What is Predictive Modeling?

Predictive modeling is a technique for forecasting the probability of a certain event taking place in the future. It entails using current and past data to forecast the future events. In predictive modeling, you use known outcomes of historical data to determine whether specific patterns are likely to recur in the future. What is Lifetime Value Modeling? Lifetime Value Modeling is a method of forecasting the long-term earnings and profit of a business. It's a strategy for calculating the cumulative amount of profit generated by a customer over the life of their relationship with a company. Lifetime Value Modeling is used to decide the most effective ways to engage with consumers, such as personalized deals or special promotions, to maximize their lifetime value to the company by encouraging them to purchase more often and spend more during each transaction.

Predictive modeling and lifetime value modeling are distinct concepts that serve different purposes. Predictive modeling is used to forecast the future occurrence of specific events, whereas lifetime value modeling is used to calculate the long-term value of a customer to a company. So, the given statement is false.

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You plan to manufacture a Product X in Cote d'Ivoire, the operating cash flow in Year 1 is $6,320.

To calculate the operating cash flow in Year 1, we need to consider the following components: revenue, cost of goods sold (COGS), fixed costs, depreciation, taxes, and changes in net working capital.

Revenue: The revenue is calculated by multiplying the number of units sold by the selling price per unit. In this case, the revenue is 8,000 units x $5 = $40,000.

COGS: The cost of goods sold is the cost per unit multiplied by the number of units sold. Here, the COGS is 8,000 units x $3 = $24,000.

Fixed Costs: The fixed costs are given as $10,000.

Depreciation: Since the equipment has a life of 3 years and was purchased for $12,000, the annual depreciation expense is $12,000/3 = $4,000.

Taxes: The tax rate is 30%. We calculate the taxable income by subtracting the COGS, fixed costs, and depreciation from the revenue: $40,000 - $24,000 - $10,000 - $4,000 = $2,000. The tax liability is then $2,000 x 30% = $600.

Changes in Net Working Capital: The change in net working capital in Year 1 is -$10,000.

Now, we can calculate the operating cash flow: Operating Cash Flow = Revenue - COGS - Fixed Costs + Depreciation - Taxes + Changes in Net Working Capital = $40,000 - $24,000 - $10,000 + $4,000 - $600 - (-$10,000) = $6,320.

Therefore, the operating cash flow in Year 1 is $6,320.

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a) Sample distribution follows normal distribution with mean( μ) = 1.10 kg,

and standard deviation σ = 0.081

b) Sample distribution follows normal distribution with mean( μ)  = 1.10 kg,

and standard deviation σ = 0.04

d) The percentage of all samples of three men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.10 kg is 79.77%.

e) The percentage of all samples of twelve men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.10 kg is 99.3%.

2) 68.26% of men in country A have brain weights between 1.48 kg and 1.72 kg.

Solution:

Population standard deviation is the measure of how spread out the population data is. It measures the difference of the individual items from the mean. A standard deviation is a statistic that measures the dispersion of a dataset relative to its mean. It is calculated as the square root of variance by determining the variation between each data point relative to the mean.

1)

Given mean = 1.10 kg, standard deviation = 0.14 kg

a) To find the sampling distribution of the sample mean for samples of size 3.

Standard error of mean = σ/√n

= 0.14/√3

=0.081

Sample distribution follows normal distribution with mean( μ) = 1.10 kg,

and standard deviation σ = 0.081

b) To find the sampling distribution of the sample mean for samples of size 12.

Standard error of mean = σ/√n

= 0.14/√12

= 0.04

Sample distribution follows normal distribution with mean( μ)  = 1.10 kg,

and

standard deviation σ = 0.04

d) Determine the percentage of all samples of three men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.10 kg.

Sample distribution follows normal distribution with mean( μ)  = 1.10 kg,

and

standard deviation σ = 0.081

Z = (x - μ) / σZ

= (1.1 + 0.1 - 1.1) / 0.081

= 1.23

Z = (1.1 - 0.1 - 1.1) / 0.081

= -1.23

P ( -1.23 < Z < 1.23) = 0.7977

The percentage of all samples of three men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.10 kg is 79.77%.

e) Determine the percentage of all samples of twelve men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.10 kg.

Sample distribution follows normal distribution with mean( μ )= 1.10 kg,

and

standard deviation σ = 0.04

Z = (x - μ) / σ

Z = (1.1 + 0.1 - 1.1) / 0.04

= 2.5

Z = (1.1 - 0.1 - 1.1) / 0.04 = -2.5

P ( -2.5 < Z < 2.5) = 0.993

The percentage of all samples of twelve men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.10 kg is 99.3%.

2)

Given mean = 1.60 kg,

standard deviation = 0.12 kg

68.26% of men in country A have brain weights between μ - σ and μ + σ

68.26% of men in country A have brain weights between 1.48 kg and 1.72 kg.

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Geometric sequence with a common ratio of 1/2. Rule for the nth term: an = 900  (1/2)^(n-1).

A sequence is considered arithmetic if the difference between consecutive terms is constant, and it is geometric if the ratio between consecutive terms is constant. In the given sequence, we can observe that each term is half of the previous term, indicating a constant ratio of 1/2.

To find the rule for the nth term of a geometric sequence, we start with the first term and multiply it by the common ratio raised to the power of (n-1), where n represents the position of the term. In this case, the first term is 900, and the common ratio is 1/2. Therefore, the rule for the nth term of the sequence is an = 900 (1/2)^(n-1).

Using this rule, we can find any term in the sequence by substituting the corresponding value of n into the formula. For example, the third term can be found by setting n = 3: a3 = 900 (1/2)^(3-1) = 225.

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Answer:

A. 223 were not playing volleyball

B. 12 books on each shelf with 3 books left over..

C. 12.15$ a day

D. 12.15 miles per day.

Step-by-step explanation:

A. Subtract 20 from 243.

B. Since books cant be divided into fractions, you take 243 divided by 20, which is 12.15. However, we have to say 12 with 3 leftover because books isn't money, a distance measure, etc. and cannot be cut into pieces.

C. 243 / 20 is 12.15

D. 243 / 20 is 12.15

Answer:

C

Step-by-step explanation:

(c + 1)(c+1) = c² + c + c + 1 = c² + 2c + 1

Answer:

1. 3

2. 1

Step-by-step explanation:

Answer:

1.3. now mark brainlieet plsssssssssss like you said you would

Answer: I think B is the answer.

Step-by-step explanation:

Answer:not sure but

Step-by-step explanation: you can multiply each box, so lets say we have 20 times 7cm you multiply that and you get your answer, so put that a side

and once you multiply in each box add the numbers up and that should get your answer and make sure to add the "cm" at the end

Answer: y-11 = -3(x+2)

Answer:

A

Step-by-step explanation:

x=

−b±√b2−4ac

2a

x=

−(2)±√(2)2−4(2)(15)

2(2)

x=

−2±√−116

4

and there is really no solution

Answer:

n/a

Step-by-step explanation:

how does it feel

The pair (6, 3) in R, (6, 4) in R, and (C, 3) in R couldn't be matched with any pair in S, so they are not included in the composition relation RS.

To find the composition relation RS of R with S, we need to combine the ordered pairs in R and S in a way that the second element of each pair in R matches the first element of the pair in S.

Let's perform the composition:

R = {(2, 1), (6, 3), (6, 4), (C, 3)}

S = {(1, y), (1, 2), (2, w), (3, z)}

To form RS, we match the second element of each pair in R with the first element of each pair in S:

(2, 1) in R matches with (1, y) in S, so we combine them to form (2, y).

(2, 1) in R matches with (1, 2) in S, so we combine them to form (2, 2).

(6, 3) in R doesn't match with any pair in S.

(6, 4) in R doesn't match with any pair in S.

(C, 3) in R doesn't match with any pair in S.

Therefore, the composition relation RS is: RS = {(2, y), (2, 2)}

The pair (6, 3) in R, (6, 4) in R, and (C, 3) in R couldn't be matched with any pair in S, so they are not included in the composition relation RS.

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Using a one-sample t-test, we cannot conclude that the mean time to spoil is significantly different when bananas are hung from the ceiling.

3.9, 4.9, 5.1, 3.9, 4, 5.8, 7, 5, 3.6, 4.3, 4.4, 6, 6.8, 6.7, 7.1, 5.2

We can calculate the sample mean and sample standard deviation:

Sample mean (x) = (3.9 + 4.9 + 5.1 + 3.9 + 4 + 5.8 + 7 + 5 + 3.6 + 4.3 + 4.4 + 6 + 6.8 + 6.7 + 7.1 + 5.2) / 16 = 5.3

Sample standard deviation (s) = √[(Σ(xi - x)²) / (n - 1)] = √[(Σ( - 5.3)²) / 15] ≈ 1.273

We will perform a one-sample t-test using the null hypothesis (H0) that the mean time to spoil is equal to 6.2 days, and the alternative hypothesis (H1) that the mean time to spoil is less than 6.2 days.

The test statistic is calculated as:

t = (x - μ) / (s / √n)

Where μ is the hypothesized mean (6.2), s is the sample standard deviation (1.273), and n is the sample size (16).

Plugging in the values:

t = (5.3 - 6.2) / (1.273 / √16) ≈ -0.887

To determine the critical t-value for a one-tailed test at α = 0.05 level of significance with 15 degrees of freedom (n - 1), we refer to the t-distribution table or use statistical software. The critical t-value is approximately -1.753.

Since the test statistic (-0.887) does not exceed the critical t-value (-1.753), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the mean time to spoil is less when bananas are hung from the ceiling compared to the average time of 6.2 days, at the α = 0.05 level of significance.

Therefore, we cannot conclude that the mean time to spoil is significantly different when bananas are hung from the ceiling.

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True. Any solution of Problem 1 (max f(x)) is also a solution of Problem 2 (max f(x) subject to g(x) = c).

True. If Problem 1 does not have a solution, then Problem 2 does not have a solution.

True. Problem 2 (max f(x) subject to g(x) = c) is equivalent to min -f(x) subject to g(x) = c.

False. In Problem 2, the quasi-convexity of f is not a sufficient condition for a point satisfying the first-order conditions to be a global minimum.

The function f(x,y) = 5x - 17y is quasi-concave.

Any solution that maximizes f(x) will also satisfy the constraint g(x) = c. Therefore, any solution of Problem 1 is also a solution of Problem 2.

If Problem 1 does not have a solution, it means that there is no maximum value for f(x). In such a case, Problem 2 cannot have a solution since there is no maximum value to subject to the constraint g(x) = c.

Problem 2 can be reformulated as finding the minimum of -f(x) subject to the constraint g(x) = c. This is because maximizing f(x) is equivalent to minimizing -f(x) since the maximum of a function is the same as the minimum of its negative.

False. Quasi-convexity of f is not a sufficient condition for a point satisfying the first-order conditions to be a global minimum in Problem 2. Quasi-convexity guarantees that local minima are also global minima, but it does not ensure that the point satisfying the first-order conditions is a global minimum.

The function f(x,y) = 5x - 17y is quasi-concave. A function is quasi-concave if the upper contour sets, which are defined by f(x,y) ≥ k for some constant k, are convex. In this case, the upper contour sets of f(x,y) = 5x - 17y are convex, satisfying the definition of quasi-concavity.

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Answer:

Step-by-step explanation:

You need to use vertex form of a quadratic to solve this.

Consider the vertex to be [tex](h,k)[/tex]

Another way of representing a quadratic is in "vertex form":

[tex]f(x) = a(x-h)^2+k[/tex]

Now all you have to do is solve for a.  You know that the vertex is [tex](0,2)[/tex] and you have know the point of [tex](1,8)[/tex].  Now, all you have to do is plug in these values and solve for a.

[tex]8 = a(1-0)^2+2\\8=a(1)^2+2\\8=a+2\\a=6[/tex]

Now you know the equation is [tex]f(x) = 6(x-0)^2+2[/tex] , but you need it in quadratic form.  All you have to do is solve is distribute the 6:

[tex]6x^2+2[/tex]

You get:

a = 6

b = 0

c = 2

Please mark this as brainliest if it satisfies your question

Answer:

the answer is A. 6 1/3 feet

Step-by-step explanation:

There are 12 inches in a foot. 12 x6=72 r of 4

there are 12 inches in a foot 12÷4=3. 1/3 of a foot

6 + 1/3= 6 1/3

Given information: The hourly number of emergency telephone calls coming in to a police command and control center has approximately a normal distribution with a mean of 130 and standard deviation of 25. One operator can deal with 24 calls in an hour.

a) The probability that 6 operators will be able to deal with all the calls that arise in an hour is 0.7642.

b) The number of operators should be 203 to ensure that there is sufficient capacity to meet 95% of demand.

c) (i) The probability that 12 operators will be able to deal with all the calls that arise in an hour is 0.7852.

(ii) The number of operators should be 336 to ensure that there is sufficient capacity to meet 95% of demand.

a) Probability that 6 operators will be able to deal with all the calls that arise in an hour.

Mean, µ = 130, Standard Deviation, σ = 25.

Operator can deal with in an hour, n = 24.

Let X = number of emergency calls coming in an hour.

The number of emergency telephone calls coming in to a police command and control center in an hour can be assumed to be Poisson with λ = 130.

Since each operator can handle 24 calls in an hour, therefore, the number of operators required to handle all the calls can be obtained as follows: [tex]$$\frac{X}{24}$$[/tex].

This can be converted to a Standard Normal Variable Z using the formula:[tex]$$Z=\frac{(\frac{X}{24}-\mu)}{\sigma}$$[/tex].

Probability that 6 operators will be able to deal with all the calls that arise in an hour can be calculated as follows:

[tex]$$\begin{aligned} \frac{X}{24} &\leq 6 \\ X &\leq 6 \times 24 \\ X &\leq 144 \end{aligned}$$[/tex]

Now, we need to find the probability of Z ≤ [tex]$$(\frac{144}{24}-130)/25=0.72$$[/tex].

Using normal distribution tables, we get P(Z ≤ 0.72) = 0.7642.

Hence, the probability that 6 operators will be able to deal with all the calls that arise in an hour is 0.7642.

b) To find the number of operators should there be to ensure that there is sufficient capacity to meet 95% of demand.

Let X = number of emergency calls coming in an hour.

The number of emergency telephone calls coming in to a police command and control center in an hour can be assumed to be Poisson with λ = 130.

Since each operator can handle 24 calls in an hour, therefore, the number of operators required to handle all the calls can be obtained as follows:[tex]$$\frac{X}{24}$$[/tex].

This can be converted to a Standard Normal Variable Z using the formula:[tex]$$Z=\frac{(\frac{X}{24}-\mu)}{\sigma}$$[/tex].

To ensure sufficient capacity to meet 95% of demand, we need to find the value of X such that: P(X ≤ x) = 0.95.

Using the Z table, we can find that the probability of Z ≤ 1.645 is 0.95.

Now, we can use the formula:

[tex]$$\frac{X}{24}-130/25=1.645$$[/tex]

[tex]$$X= 1.645\times 25\times 24+130$$[/tex]

[tex]$$X=202.63$$[/tex]

Therefore, the number of operators should be 203 to ensure that there is sufficient capacity to meet 95% of demand.

c) Two centers are combined and let X_1 and X_2 be the number of calls at centers 1 and 2, respectively.

Then the total number of calls, X = X_1 + X_2, follows a normal distribution with

mean = 130 + 130

mean = 260, and

standard deviation = sqrt(25^2 + 25^2)

= 35.36

i) Probability that 12 operators will be able to deal with all the calls that arise in an hour can be calculated as follows:

[tex]$$\begin{aligned} \frac{X}{24} &\leq 12 \\ X &\leq 12 \times 24 \\ X &\leq 288 \end{aligned}$$[/tex]

Now, we need to find the probability of Z ≤ [tex]$$(\frac{288}{24}-260)/35.36=0.789$$[/tex].

Using normal distribution tables, we get P(Z ≤ 0.789) = 0.7852.

Hence, the probability that 12 operators will be able to deal with all the calls that arise in an hour is 0.7852.

ii) To ensure sufficient capacity to meet 95% of demand, we need to find the value of X such that: P(X ≤ x) = 0.95.

Using the Z table, we can find that the probability of Z ≤ 1.645 is 0.95.

Now, we can use the formula:

[tex]$$\frac{X}{24}-260/35.36=1.645$$[/tex]

[tex]$$X= 1.645\times 35.36\times 24+260$$[/tex]

[tex]$$X=335.58$$[/tex]

Therefore, the number of operators should be 336 to ensure that there is sufficient capacity to meet 95% of demand.

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Answer: d. y= -2 2x+y= -3

Step-by-step explanation: Hope this help :D

Answer

672 meters²

Step-by-step explanation:

2×(18×12 + 18×4 + 12×4) = 672 meters²

hope this helps :))

Given a fixed vector b and a vector x is a solution to Qx = b, it is required to prove that every solution to the equation is in the form x = xh + xp where xh is a particular solution to Qx = b and xp is a solution to the equation Qxp = 0.

Let xh be a particular solution to Qx = b, so that Qxh = b.

Now consider the homogeneous equation Qx = 0.

This is an m × n system of homogeneous linear equations in the n unknowns x1, x2, ..., xn, whose coefficient matrix is Q.

Since xh is a solution to the equation Qx = b, it follows that the equation Q(x - xh) = Qx - Qxh = b - b = 0.

This means that x - xh is a solution to the homogeneous equation Qx = 0.

Now any solution to Qx = b is of the form x = xh + xp, where xp is any solution to the homogeneous equation Qxp = 0.

Thus, every solution to the equation is in the form x = xh + xp, as required.

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