What is the nature of roots of the quadratic equation 9x2 6x 2 0?

(i) 0 is the remainder when x³ + 3x² + 3x + 1 is divided by (x +1)

(ii) 27/8 is the remainder when x³ + 3x² + 3x + 1 is divided by (x - 1/2)

(iii) 1 is the remainder when x³ + 3x² + 3x + 1 is divided by x

(iv) -π³ + 3π² - 3π + 1 is the remainder when x³ + 3x² + 3x + 1 is divided by (x + π)

(v) -27/8 is the remainder when x³ + 3x² + 3x + 1 is divided by (5 + 2x)

The value remaining after division is known as the Remainder. After division, we are left with a value if a number (dividend) cannot be divided entirely by another number (divisor). The remaining is the name for this amount.

Let a be any real number and let p(x) be any polynomial with degree greater than or equal to one.

Now, if a polynomial p(x) is divided by (x - a) then the remainder is p(a).

Assume, p(x) = x³ + 3x² + 3x + 1

Now,

(i) The root of x + 1 = 0 is -1

or, p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

or, p(-1) = -1 + 3 - 3 + 1

or, p(-1) = 0

Hence by the remainder theorem, 0 is the remainder when, x³ + 3x² + 3x + 1 is divided by x + 1. We can also say that x + 1 is a factor of x³ + 3x² + 3x + 1.

(ii) The root of x - (1/2) = 0 is 1/2.

or, p(1/2) = (1/2)³ + 3(1/2)² + 3(1/2) + 1

or, p(1/2) = 1/8 + 3/4 + 3/2 + 1

or, p(1/2) = (1 + 6 + 12 + 8)/8

or, p(1/2) = 27/8

Hence by the remainder theorem, 27 / 8 is the remainder when x³ + 3x² + 3x + 1 is divided by x.

(iii) The root of x = 0 is 0

or, p(0) = (0)³ + 3(0)² + 3(0) +1

or, p(0) = 0 + 0 + 0 + 1

or, p(0) = 1

Hence by the remainder theorem, 1 is the remainder when  x³ + 3x² + 3x + 1 is divided by x.

(iv) The root of x + π = 0 is - π

or, p(-π) = (-π)³ + 3(π)² - 3π + 1

or, p(-π) = -π³ + 3π² - 3π + 1

Hence by the remainder theorem, (-π)³ + 3(π)² - 3π + 1 is the remainder when x³ + 3x² + 3x + 1 is divided by x + π.

(v) Now, the root of 5 + 2x = 0 is -5/2

or, p(-5/2) = [(-5/2)³ + 3(-5/2)² + 3(-5/2) + 1]

or, p(-5/2) = [(-125/8) + (75/4) + (-15/2) + 1]

or, p(-5/2) = (-125 + 150 - 60 + 8) / 8

or, p(-5/2) = (-185 + 158) / 8

or, p(-5/2) = -27/8

Hence by remainder theorem, -27/8 is the remainder when x³ + 3x² + 3x + 1 is divided by 5 + 2x.

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The complete question is as follows:

Find the remainder when x³ + 3x² + 3x + 1 is divided by

i) x + 1    ii) x - (1/2)    iii) x    iv) x + π    v) 5 + 2x

The graph of the function is illustrated below and the domain of the function is [-2, 2].

The term domain in math is referred as the set of all inputs of the function

Here we have to find and sketch the domain of the function f(x, y, z) = ln(64 − 4x² − 16y² − z²)

In order to sketch the function, we have to use the graphing calculator, and following the listed steps to sketch the graph.

The first step is to label the graph with appropriate axis like x, y, and z.

Now we have to input the function f(x, y, z) = ln(64 − 4x² − 16y² − z²) on the input field and then click on generate.

Now, we get the graph for the function.

After that we have to adjust the limit for the given graph.

Then we finally get the resulting graph as follows.

As per the definition of domain, and with the help of the given graph, we have identified the value of the domain is [-2, 2].

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The required property that will change with the addition of values in the set will change the mean, median, and mode, While multiplication changes the standard deviation and also the mean by constant.

Statistics is the study of mathematics that deals with relations between comprehensive data.

Each of the statistic properties will change by adding the number to each number in the set, except properties such as standard deviation and variance. While multiplying the set element with a number each and every property of the statistic corresponding to the set will change.

Thus, Options A, B, and D are correct.

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It is an iterated function also known as hailstone problem and can be solved on iterated steps.

Since then, the 3X + 1 issue has taken many different shapes. It is among the most infamous puzzles that have never been solved. For more than 40 years, prizes have been offered for its resolution, but nobody has totally and properly done so.

The 3x+1 Conjecture states that this function can be iterated through multiple times to eventually produce the value 1 starting from any positive integer n.

Take a positive integer X and divide it by 2 if it is even. If it is odd, multiply it by 3 and add 1 to the result to form the conjecture 3X + 1. The ends operate repeatedly according to the aforementioned criteria, and after a finite number of repetitions, the final end is always 1.

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Answer: Part A: 994 feet higher thanks to absolute value.

Part B: -86 feet.

Step-by-step explanation: math

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion. the missing lengths of HF is 4 and missing length PR =84.

What are Similar Triangles?

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion .

Triangle QRS is similar to triangle FGH.

Let us find the length of unknown side.

By comparing the sides.

SR/QR=HF/HG

8/14 = x/7

Apply cross multiplication

14x = 56

Divide both sides by 14

x = 56/14

x = 4

HF = 4

8)

By properties of triangle we have

QP/BP=PR/PC

56/24= x/36

Apply cross multiplication

24x = 2016

Divide both sides by 24

x = 2016/24

x = 84

PR = 84

Hence the missing lengths of HF is 4 and missing length PR =84.

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Answer:

Step-by-step explain

∛12↔3³≈4x↑

For solving algebra Determine the issue, Describe your knowledge, Plan beforehand, Execute the plan ,and Check to see whether the response makes sense.

One of the many different mathematical disciplines is algebra. Algebra, a common thread that runs through nearly all of mathematics, is, roughly speaking, the study of mathematical symbols and the rules for using these symbols in formulae. With the use of mathematical expressions, algebra is the area of mathematics that helps to depict situations or problems. We employ integers that have a fixed or definite value in algebra, such as 2, 7, 0.068, etc. In addition to using numbers, algebra also makes use of variables like x, y, and z.

to do algebra for beginners - Determine the issue, specify your knowledge, Plan, execute, and confirm that the solution makes sense.

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Constant, variable, operator and terms are four types of expression.

A constant term in mathematics is a term in an algebraic expression whose value is fixed or cannot change since it lacks any variables that may be changed. As the component is not yet included in the new term, a constant term that has a constant applied as a multiplicative coefficient likewise indicates a constant term. The term identify themselves as constants because the phrase has been modified. 5 is a constant term, in the quadratic polynomial 2x^2 + 5.

Yes, negative sign have nothing to do with the definition of constant. -4 is still a constant.

There are four types of expression in mathematics:

1. Constant: any numeric value e.g. 6

2. Variable: any alphabetic/unknow value e.g. x,y etc.

3. Operators: +,-, *, / (addition, subtraction, division, multiplication)

4. Term: it can be constant, variable, constant multiplied by variable. It is separated by an operator.

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Answer:

Step-by-step explanation:

it is actually in the textbook pg 119

Answer:

D

Step-by-step explanation:

It can't be A or B because those graphs are positive, and our graph is negative.

It can't be C because there is a point at (2,-5), but we don't see that point on this graph!

So, the answer is D

Answer:

d

Step-by-step explanation:

f(x) = \sqrt{x+2} - 5

The probability of selecting a number greater than 19 from 1 to 25 is 6/25 or 0.24.

The probabilities of an event occurring are defined by probability.

The ratio of favorable outcomes to all possible outcomes of an event is known as probability.

The formula for the Probability of an event is given by

Probability(Event) = Favorable Outcomes/Total Outcomes

Here we have

The set of numbers is from 1 to 25

Total number of numbers from 1 to 25 = 25

In given set of numbers, the number which is greater than 19 are

=> 20, 21, 22, 23, 24, and 25.

The number of favorable outcomes (getting a number, > 19 ) = 6

From the above information,

Probability of selecting a number greater than 19 = 6/25 = 0.24

Therefore,

The probability of selecting a number greater than 19 from 1 to 25 is 6/25 or 0.24

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The gradient of the line whose equation is x = -4 is: undefined.

The graph of an vertical line is a line that cuts the x-axis at (a, 0), of which has an undefined slope because it has a change in y but has no change in x. That is, the run of the line is zero. This means the gradient of a vertical line is undefined.

The equation of a vertical line is given as x = a, where a is the point that the line intersects the x-axis at.

Given the equation, x = -4, it means the line is a vertical line.

Therefore, since the gradient of vertical lines are undefined, the gradient of x = -4 is also determined to be undefined.

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The commission the employee earns as a percentage of the total cost of the gym membership is 19%.

A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100. The percentage therefore refers to a component per hundred. Per 100 is what the word percent means. It is represented by %.

In this case, the gym keeps $42 and the employee who sold it gets $8.

The commission the employee earns as a percentage of the total cost of the gym membership will be:

= 8 / 42 × 100

= 19%

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A slope of 5 indicates that the corresponding y varies by 5 units for every unit change in x

The slope or gradient of a line in mathematics is a numerical representation of the steepness and direction of the line.

Given:

The slope of 5.

Think of a slope of 5, which entails climbing 5 steps for each step you take. Much more quickly, you ascend to the top.

The slope is a measurement that indicates how much y shifts when x shifts.

For instance, a slope of 5 indicates that the corresponding y varies by 5 units for every unit change in x.

If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,

then the equation of line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

Therefore, think of a slope of 5, which entails climbing 5 steps for each step you take. Much more quickly, you ascend to the top.

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The inverse of f is a function which takes any value x and divides it by 4, then adds 7 to the result.

The inverse of a function is the opposite of the original function, meaning that it takes the output of the original function and turns it back into the input. In this case, the original function is f(x)=1/4x+7. To find the inverse, we need to solve for x in the equation, which can be done by subtracting 7 from both sides and then multiplying both sides by 4. This gives us x=(x+7)/4, which is the inverse of the original function. To put it simply, the inverse of f takes any output of the original function, adds 7 to it, and then divides it by 4 to get the original input.

example;

If f(x)=1/4x+7 and x=3, then

f(3) = (1/4)3 + 7

7.75

The inverse of f is f^-1(x) = (x + 7) / 4.

If f^-1(7.75) = (7.75 + 7) / 4, then

f^-1(7.75) = 14.75 / 4

3.6875

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Answer:

3

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7 cm}\underline{Scalar projection}\\\\Scalar projection $=\dfrac{u \cdot v}{|u|}$\\\\where:\\ \phantom{ww}$\bullet$ $u$ is the vector being projected onto.\\\end{minipage}}[/tex]

Given:

Calculate the magnitude of vector u:

[tex]\implies |u|=\sqrt{(0)^2+(-1)^2}=1[/tex]

The scalar projection is the magnitude of the vector projection.

Therefore, the magnitude of the projection of vector v onto vector u is:

[tex]\implies \dfrac{u \cdot v}{|u|}=\dfrac{\langle0, -1\rangle \cdot \langle-5, -3\rangle}{1}=\dfrac{0 +3}{1}=\dfrac{3}{1}=3[/tex]

The slope of the given equation y = 10 - 4x is equal to -4.

As given in the question,

Given equation is equal to y = 10 - 4x

Standard form in slope intercept form for the equation  is given by :

y = mx  + c

'm' is representing the slope of the given equation

'c' is representing the value of the  y-intercept of the given equation.

Now after simplifying the given equation in the standard form we get,

y = 10 - 4x

⇒ y = -4x + 10 __(1)

Compare equation (1) with the standard form of slope intercept form we get,

Slope of the equation is equal to m = -2.

Therefore, the slope of the given equation y = 10 - 4x is given by m = -4.

The given question is incomplete , the complete question is:

What is the slope of the equation y = 10 - 4x ?

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Answer:

Step-by-step explanation:

The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by ɡ0 or ɡn, is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2 (about 32.17405 ft/s2).

Answer:

3/7

Step-by-step explanation:

[tex]g(-2)=\log_{2}(1)-4=-4 \\ \\ g(5)=\log_{2}(8)-4=-1 \\ \\ \frac{g(5)-g(-2)}{5-(-2)}=\frac{-1-(-4)}{5-(-2)}=\frac{3}{7}[/tex]

The name of the condition in which factors f and g are correlated and may not be independent is confounded

What is  Fractional factorial?

In statistics, fractional factorial designs are experimental layouts made up of a carefully selected subset of full factorial design's experimental runs.

In a fractional factorial experiment, confounded factors are those that cannot be varied independently of each other because they are correlated. Confounded factors can affect the interpretation of the results of the experiment, as it may not be clear which factor is responsible for a particular effect. To control for confounded factors, it may be necessary to include them both in the experimental design or to use additional statistical techniques to account for their correlated nature.

Confounded factors are also known as aliased factors.

Hence, Name of the condition in which factors f and g are correlated and may not be independent is confounded

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Given a system of two equations in two variables, solve using the substitution method.

Word problems that require us to write systems of linear equations have two unknown quantities and two different ways to relate them.

This means we need to write two linear equations, and each contains the two unknowns as variables. The Understanding linear relationships lesson details how to translate verbal descriptions into equations.

After we write the equations, we can solve the system using our preferred method(s) for solving systems of linear equations covered in the Solving systems of linear equations lesson.

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The two triangles are similar if all three sides of one triangle are in proportion to the three sides of another triangle.

SSS, which stands for "side, side, side," denotes that we have two triangles with the same number of sides on each pair of corresponding sides. Two triangles are comparable if they have three pairs of sides that are in the same ratio. According to the SSS test for triangle similarity, two triangles are considered similar if their third sides are proportionate to one another.

In our daily lives, the idea of comparable triangles is really useful. By measuring the shadows' lengths and then utilizing triangles that are comparable to them, we can determine an object's height, such as the height of a building or a tower. Compare the shortest sides first, followed by the longest sides, when applying the SSS Similarity Theorem. Two triangles are similar if their corresponding side lengths are proportionate to one another.

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Answer:

4

Step-by-step explanation:

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 x 2 = 4. The square root of 9 is 3, because 3 x 3 = 9.

The square root of 16 is 4, because 4 x 4 = 16. Therefore, the value of the square root of 16 is 4.

Answer:

Step-by-step explanation:

To find the area of the base of the box, we need to determine the length, breadth, and height of the box. We can set up a system of equations to represent the given information:

breadth = 2/3 * length

height = 1/3 * length

volume = length * breadth * height

Substituting the second equation into the third equation, we get:

volume = length * breadth * (1/3 * length)

      = length^2 * (2/3) * (1/3)

      = (2/9) * length^2

      = 48

      = 2^4 * 3

Solving for length, we find that length = 6. Plugging this value back into the first equation, we find that breadth = 4.

Therefore, the area of the base of the box is length * breadth = 6 * 4 = 24 square meters.

The measure of each side of the isosceles right triangle with hypotnuse 2.5√2 cm is  2.5 cm   .

The Isosceles Right Triangle is a type of right triangle which consists of two equal length legs .

In the right isosceles triangle the measure of the base and the perpendicular is same , So , base = perpendicular

let the side of the isosceles right triangle be = x .

given that the measure of the hypotnuse is = 2.5√2 cm

In a right triangle , By Pythagoras Theorem ,

base + perpendicular = hypotnuse

x² + x² = (2.5√2)²

2x² = 6.25 × 2

2x² = 6.25 × 2

2x² = 12.5

x² = 12.5/2

x² = 6.25 ,

x =  2.5 cm   ,....by taking square root on both the sides .

Therefore , the measure of each side is  2.5 cm  .

The given question is incomplete , the complete question is

What is the measure of each side of the given isosceles right triangle with a hypotenuse that measure 2.5√2 cm ?

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The result of the each of the following set is

A ∪ B = {1, 3, 5, 6, 7, 9}

A ∩ B = {3, 9}

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A ∩ C = {∅}

A - B = {1, 5, 7}

B - A = {6}

B U C = {2, 3, 5, 6, 8, 9 }

B ∩ C = {6}

The given values are

A = {1, 3, 5, 7, 9}

B = {3, 6, 9}

C = {2, 4, 6, 8}

Then find the each given terms in set roaster notation

Union of the set, intersection of the set  and the difference of the set are the basic operations of set

A ∪ B = {1, 3, 5, 6, 7, 9}

A ∩ B = {3, 9}

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A ∩ C = {∅}

A - B = {1, 5, 7}

B - A = {6}

B U C = {2, 3, 5, 6, 8, 9 }

B ∩ C = {6}

Therefore, all the given terms has been found

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We will use the concept of Simplification of Equations . The value of p in the given equation is 1.

An Algerbraic equation in algebra is any equation that can be transformed in standard form like the following:

ax+bx+c=0 or ap+bp+cp+d=0 it can be of any form.

With a 0 and a, b, and c are known numbers, x is an unknown value, and a. The equation is linear, not quadratic, if a = 0 and b 0. The coefficients of the equation, denoted by the numbers a, b, and c, are the quadratic coefficient, linear coefficient, and constant or free term, respectively.

The roots or zeros of the expression on the left-hand side of the equation are the values of x that fulfill the equation. There can only be only solution to an Algebraic equation.

6p=2(5-2p) is the given equation

Divide by 2 on both LHS and RHS of the equation

6p/2 = 2(5-2p)/2

⇒3p=5-2p

⇒5p=5

∴p=1

We will use the concept of Simplification of Equations . The value of p in the given equation is 1.

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Answer: 56 – 41 = 15

Step-by-step explanation: