For what values of k the roots of the quadratic equation 9x 2 )+ 3kx 4 0 are equal?

Answer:

[tex]\displaystyle{\theta=\dfrac{5\pi}{4}\, , \, \dfrac{9\pi}{4}}[/tex]

Step-by-step explanation:

Given the equation [tex]\displaystyle{2\cos \theta = 2\sin \theta}[/tex]. We have to divide both sides by 2 which gives us [tex]\displaystyle{\cos \theta = \sin \theta}[/tex]. Then divide both sides by [tex]\displaystyle \cos \theta[/tex] which gives us [tex]\displaystyle{1=\dfrac{\sin \theta}{\cos \theta}}[/tex].

We know that:

[tex]\displaystyle{\dfrac{\sin \theta}{\cos \theta} = \tan \theta}[/tex]

So we can rewrite the equation as [tex]\displaystyle{1=\tan \theta}[/tex].

We know that [tex]\displaystyle{\tan \theta = 1}[/tex] when [tex]\displaystyle{\theta = \dfrac{\pi}{4}+\pi k}[/tex] for [tex]\displaystyle{k \in I}[/tex] (k is any integer). However, the equation is given with the interval [tex]\displaystyle{\pi \leq \theta \leq 3\pi}[/tex]. Therefore, we have to substitute k-values that satisfy the interval.

It appears that only k = 1, 2 which gives us [tex]\displaystyle{\theta_1 = \dfrac{\pi}{4}+\pi}[/tex] and [tex]\displaystyle{\theta_2 = \dfrac{\pi}{4}+2\pi}[/tex]can be used since both satisfies both values and interval.

Simplifying both solutions:

[tex]\displaystyle{\theta_1 = \dfrac{\pi}{4}+\dfrac{4\pi}{4} \, , \, \theta_2=\dfrac{\pi}{4}+\dfrac{8\pi}{4}}\\\\\displaystyle{\theta=\dfrac{5\pi}{4}\, , \, \dfrac{9\pi}{4}}[/tex]

The simulation disproves the hypothesis that 1 out of 10 players are arbitrarily chosen to be spies by showing that being selected as a spy in 5 out of 7 games is improbable.

A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%.

The probability is a metric for determining how likely an event is to occur. It gauges the event's degree of certainty. The probability formula is presented as follows:

P(E) = Number of Favorable Outcomes/Number of Total Outcomes.

The frequency is o when the number of agnes chosen to be soy is 5, as shown in the table.

The simulation demonstrates that getting chosen as a spy in 5 out of 7 games is implausible, refuting the claim that 1 out of 10 players is randomly chosen to be a spy.

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When solving a linear system of equations, what you obtain are the roots of the equation.

A linear equation is the type of equation in which he highest power of the variable is one. In some cases, we would have a system of linear equations and we have to solve the system.

When we are solving the system of equations, there are various methods that we could use such as;

The solutions that we obtain from the equation are the root of that equation.

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On solving the provided question we can say that, - the roots of the given equations [tex]3x^2 + 7x - 6[/tex] are - x2/3 and x = -3

When referring to the values of a variable for which the supplied polynomial equals zero, we use the term "roots of a polynomial." The polynomial p(x) has zero roots if and only if an is its root. By setting each component to 0, you may solve for x to determine the roots, or solutions, of the polynomial equation P(x) = 0. Factoring is used to solve the polynomial equation. Set each component to a value of 0 (2x4), (x-6) or (x+1). Calculate x using the solution.

here,

[tex]3x^2 + 7x - 6[/tex]

(x - 2/3)(x+3)

x = 2/3 and x = -3

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All equilateral triangles are NOT congruent.

Only those equilateral triangles whose sides have the same length will be congruent.

Those whose sides have different lengths will not be congruent.

There are 60, odd numbers of 3 digits are there which have all 3 digits distinct with each of the digits odd.

we know that,

Now there are 5 odd numbers from 0 to 9. Those are 1,3,5,7 and 9. Now we need to fill these into three places and make sure that the numbers do not repeat.

Let us take the example of the number 135.

1st digit  2nd digit    3 digit

1               3          5

Now 135 is the first number which satisfies all the conditions.

Similarly, we can find many numbers but this would take a lot of time.

Let us generalize this.

Now we have 5 options to be put in the 1st digit place. So

1st place = 5 options.

Now since the numbers do not have to repeat the number of options we need to put in the second place will reduce to (5-1) that is 4. Thus, 2nd place = 4 options.

Now in the third place too the numbers should not repeat. Thus our options further reduce to (5-3) that is 3 options. Thus,

3rd place = 3 options.

Now, to find all the arrangements, we multiply the number of options in all places.

Thus the total number of numbers is equal to 5×4×3=60

Thus , there are 60, odd numbers of 3 digits are there which have all 3 digits distinct with each of the digits odd.

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We can solve for x and y in two equations by using the Elimination method for the simultaneous linear equations.

The Elimination approach results in the reduction of one problem to a single-variable equation when two simultaneous linear equations are solved. Following this, the answer is the same as when one line was vertical or parallel. The Gaussian elimination method is the name given to this approach.

We can understand the above method through an example:

Equation 1:     4x + 6y = 8

Equation 2:     3x + 4y = 5

Step 1: In order to ensure that the two equations have the same leading coefficient, multiply each equation by a reasonable value. Equation 1 may be easily solved by multiplying it by 3, the x coefficient from Equation 2, and Equation 2 by 4, the x coefficient from Equation 1.

Therefore the equations become,

12x + 18y = 24

12x + 16y = 10

Step 2: Subtract equation 2 from equation 1

We get 2y = 14

From the above equation, we get the value of y as 7

Step 3: We will substitute the value of y in one of the initial equations

Substituting the value of y in equation (1) 4x + 6y = 8

4x + 6(7) = 8

4x + 42 = 8

4x = 34

x = [tex]\frac{34}{4}[/tex]

x = [tex]\frac{17}{2}[/tex]

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Quadrilateral ABCD with diagonal AEC bisecting DAB, BE, DE, and 1 = 2 then BC=DC.

A quadrilateral is a plane figure that has four sides or edges, and also has four corners or vertices.

Given that Quadrilateral is ABCD.

diagonal AEC bisecting DAB,

BE, DE, and 1 = 2.

It is clearly given that the diagonals bisects each other.

Which means all the sides of the quadrilateral might be equal which means the quadrilateral may be a square or an rhombus.

Diagonals DAB and AEC are bisecting.

Then the sides BC and DC are equal.

BC=DC.

Hence, if Quadrilateral ABCD with diagonal AEC bisecting DAB,

BE, DE, and 1 = 2 then BC=DC.

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The solution for the system of equations is (-19,55).

As given the equations in the question

y = –3x – 2

Simplify the above

y + 3x = -2

5x + 2y = 15

Multiply y + 3x = -2 by 2 and subtracted from 5x + 2y = 15.

2y - 2y + 5x -6x = 15 + 4

-x = 19

x = -19

Putting the value of x in the equation y + 3x = -2.

y + 3 × - 19 = -2

y - 57 = -2

y = -2 + 57

y = 55

Therefore the solution for a system of equations is (-19,55).

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Work Shown:

1 piece = 1.75 m

11 pieces = 11*1.75 = 19.25 m

22 m - 19.25 m = 2.75 meters

This ignores the width of the saw blade, while small, it's not infinitely thin.

Step 1: Given the functions f(x) and g(x) , compute the limit limx→∞f(x)g(x) lim x → ∞ f ( x ) g ( x ) .

Step 2: If the limit in Step 1 is a finite constant a≠0 a ≠ 0 , then the growth rate of f(x) is a times the growth rate of g(x) at sufficiently large x .

Now, According to the question:

Steps on How to Compare the Rates of Change of Two Functions Using Limits

Step 1: Given the functions f(x) and g(x), compute the limit

limx→∞f(x)/g(x).

Step 2: If the limit in Step 1 is a finite constant a≠0, then the growth rate of f(x) is a times the growth rate of g(x) at sufficiently large x. If the limit in Step 1 is ∞, then the growth rate of f(x) is greater than the growth rate of g(x). If the limit in Step 1 is 0, then the growth rate of g(x) is greater than the growth rate of f(x).

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The ASA criterion might be the most appropriate, as it states that two triangles are congruent if two angles and the included side are congruent.

To use a congruence criterion in the given situation, you will need to determine which criterion is most appropriate based on the information provided. There are several congruence criteria that can be used to determine whether two triangles are congruent, including the Side-Side-Side (SSS) criterion, the Side-Angle-Side (SAS) criterion, the Angle-Side-Angle (ASA) criterion, and the Angle-Angle-Side (AAS) criterion.

In this case, the ASA criterion might be the most appropriate, as it states that two triangles are congruent if two angles and the included side are congruent. Given that ∠A=∠C=90∘ and AE=CB, you can use the ASA criterion to conclude that triangles AEB and CED are congruent.

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The question is -

Which congruence theorem can be used to prove EB = BD, AE = CB, ∠ A = ∠ C = 90°?

Distance between the points (-3, 5, -7) and (2, -4, 6) is 5√11 units.

What is cartesian coordinate points?

The Cartesian coordinates of a point in three dimensions are the triple (x, y, z). The three numbers or coordinates give the signed distance from the origin along the x, y, and z axes respectively.

Solution:

Given, x1 = -3     ,  x2 = 2

           y1 = 5      ,  y2 = -4

           z1 = -7     ,  z2 = 6

The formula for finding the distance between two points on 3D graph is given below

Distance =√((x2-x1)²+(y2-y1)²+(z2-z1)²)

⇒Distance =√{(2-(-3))²+(-4-5)²+(6-(-7))²}

⇒Distance =√{(2+3)²+(-9)²+(6+7)²}

⇒Distance =√{5²+81+13²}

⇒Distance =√(25+81+169)

⇒Distance =√275

⇒Distance =5√11

Thus, the distance between (-3, 5, -7) and (2, -4, 6) is 5√11.

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According to the given information the distance between the points 16.58 units.

the algebraic phrase known as the distance formula, which provides the distances between two places in terms of respective dimensions (see coordinate system). The distance formulae for locations in rectangular coordinates in two- and two half Euclidean space are based upon that Pythagorean theorem.

We can determine an object's actual size by learning how far it is from us. The area that an object occupies in the sky may be measured. The distance to an object is then necessary to calculate its real size. An thing appears smaller the farther it is away.

The distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂)

[tex]\begin{equation}d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}\end[/tex]

x₁, y₁, z₁ = -3, 5, -7

x₂, y₂, z₂ = 2, -4, 6

[tex]\begin{equation}d=\sqrt{\left(2+3\right)^2+\left(-4-5\right)^2+\left(6+7\right)^2}\\\begin{equation}d=\sqrt{\left(5\right)^2+\left(-9\right)^2+\left(13\right)^2}\\\begin{equation}d=\sqrt{25+81+169}\\\begin{equation}d=\sqrt{275}\\[/tex]

= 16.58 units

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1.  Mr. Hamilton wants to fill with pictures is 4 feet.

2. 6 inches is 0.5 feet.

3. Mr. Hamilton uses 8 pictures to cover to space.

4. If the pictures were 10 inches wide, then he can cover the wall by 3 pictures.

What are the units of length?

Distance is measured in length. Length has the dimension of distance in the International System of Quantities. The majority of measurement systems choose a base unit for length from which all other units are derived. The metre serves as the foundational unit of length in the International System of Units.

Meter, centimeter, kilometer, millimeter, decimeter, decameter, and other units of measurement are frequently used to describe length.

1.

The length of wall is 9.75 feet.

The length of window is [tex]5\frac{3}{4}[/tex] feet = [(5×4) + 3]/4 feet = 23/4 feet = 5.75.

The empty space of the wall is (9.75 - 5.75) = 4feet.

2.

1 feet = 12 inches

or, 12 inches = 1 feet

1 inches = 1/12 feet

6 inches = 6/12 feet =0.5 feet

3.

The number of picture is

= The length of empty space/ the wide of image

= 4/0.5

= 8

4.

Suppose the length of image 10 inches.

10 inches = 10/12 feet

The number of images is (10/12) × 4 = 10/3 = 3.333

He can not cover all empty space by using pictures. He can use only 3 pictures.

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The height of rectangle is equal to its breadth or vertical side length.

A figure which has four straight sides along with 4 right angles is called rectangle.  Two equal side lengths are parallel to each other, and the bigger sides are called length of rectangle and smaller sides are called breadth.

we can determine the height of rectangle by the following formula

if a rectangle has length, A and breadth, B

we know the area of rectangle is the product of length and breadth.

Area =A×B

we know the position of A is along the horizontal and position of B is along the vertical. Therefore, breadth, B refers the height of rectangle.

if we know the value of area and length we will find the height.

breadth = Area/length

height = Area/length

hence, the height of rectangle is its vertical side length.

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5   is the mean of the numbers .

What are the mean, median, and example?

A data collection is ordered from least to largest, and the median is the midpoint number. A data set's mode is the number that appears the most frequently. The most frequent number, or the one that happens the most frequently, is known as the mode.

                     Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.

 x = 2 + 4 + 6 + 8/4

 x = 20/4

  x = 5

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Therefore , The HCF is the divisor if the residual is 0, otherwise, apply the lemma to b and r and continue dividing until the remaining is 0.

It is possible to evaluate HCF using more than one number. It is the most effective divisor for any couple of integer that may evenly or completely divide the inputted numbers. For instance, 15 is the highest common factor between 60 and 75 since it is the largest number that divides both 60 and 75 exactly.

Here,

To calculate the HCF of the supplied numbers, many HCF formulas are utilized. See what they are now:

Product of the Numbers = LCM * HCF If P and Q are two distinct numbers, then P Q is the result of LCM (P & Q) HCF (P & Q).

Coprime numbers' HCF is always 1. LCM of Co-Prime Numbers, thus, equals the product of the numbers.

Division of Euclid To determine the HCF of two numbers, apply the lemma. Obtain the two numbers a and b such that a > b. Locate a/b , for example. The HCF is the divisor if the residual is 0, otherwise, apply the lemma to b and r and continue dividing until the remaining is 0.

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The value of x³ 3y²x² is 27y² when x is 3. This is because x³ is equal to 27, and multiplying it by 3y² and x² gives 27y².

The value of x³ 3y²x² when x is 3 can be calculated by first understanding how x³ is equal to 27. This is because when something is cubed, the value is three times itself. For example, if x is 2, then x³ is 2 x 2 x 2, which is 8. If x is 3, then x³ is 3 x 3 x 3, which is 27. Once this is understood, the value of x³ 3y²x² when x is 3 can be calculated by multiplying 27 (the value of x³ when x is 3) by 3y² and x². In this case, x² is also equal to 3, so the equation becomes 27 x 3y² x 3, which is equal to 27y². This means that the value of x³ 3y²x² when x is 3 is 27y².

x³ = 3 x 3 x 3 = 27

x³ 3y²x² = 27 x 3y² x 3 = 27y²

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The formula z = (x-μ)/σ allows us to determine the z-score for a sample mean on the distribution of means.

A Z-score is a metric that quantifies how closely a value relates to the mean of a set of values.

Standard deviations from the mean are used to measure Z-score.

A Z-score of zero means the data point's score is the same as the mean score.

Z-scores are calculated using the formula z = (x-μ)/σ, where x represents the raw score, the population mean, and the population standard deviation.

The z-score is simply the raw scoreless the population means, divided by the population standard deviation, as the calculation demonstrates.

Therefore, the formula z = (x-μ)/σ allows us to determine the z-score for a sample mean on the distribution of means.

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If the frame has a 400 square inch surface area, its breadth is 5 inches.

Let's call the width of the frame w.

The total width of the painting and the frame is 24 inches + 2w inches since there is a frame of width w on both sides of the painting.

Similarly, the total height of the painting and the frame is 18 inches + 2w inches, since there is a frame of width w on both the top and bottom of the painting.

The area of the frame is the difference between the area of the rectangular frame and the area of the painting. The area of the rectangular frame is (24 inches + 2w inches) * (18 inches + 2w inches) = 432 inches + 72w inches + 36w inches + 4w^2 inches. The area of the painting is 24 inches * 18 inches = 432 inches.

Therefore, the area of the frame is 432 inches + 72w inches + 36w inches + 4w^2 inches - 432 inches

= 72w inches + 36w inches + [tex]4w^2[/tex] inches = 400 inches.

Combining like terms, we get [tex]4w^2[/tex] inches + 108w inches = 400 inches.

Dividing both sides by 4, we get [tex]4w^2[/tex] inches + 27w inches = 100 inches.

We can use the quadratic formula to solve this equation in the quadratic form:

w = [tex]\frac{(-b +/- sqrt(b^2 - 4ac))}{2a}[/tex]

= [tex]\frac{(-27 +/- sqrt(27^2 - 4 * 1 * 100))}{2*1} \\[/tex]

= [tex]\frac{(-27 +/- sqrt(729 - 400))}{2}[/tex]

= [tex]\frac{(-27 +/- sqrt(329)) }{2}[/tex]

= [tex]\frac{(-27 +/- sqrt(289))}{2}[/tex]

The solutions are w =[tex]\frac{ (-27 + 17)}{2}[/tex]  = -5 inches and w = [tex]\frac{(-27 - 17)}{2}[/tex] = -22 inches.

Since the width of the frame must be positive, the correct solution is w = -5 inches. The frame is five inches wide.

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Answer:

-28

Step-by-step explanation:

b. The probability of drawing a red card, placing it back in the deck, and drawing another red card is approximately 22%.

This is because there are 26 red cards in a standard deck of 52 cards, out of which half (13) are red. So, when Eric draws a red card and puts it back in the deck, the probability of drawing another red card is 13/52, which is approximately 25%.

However, since the deck is shuffled after the first draw, this probability is slightly reduced to 22%. Thus, the probability of drawing a red card, placing it back in the deck, and drawing another red card is approximately 22%.

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22 1/2 because if you see the 150 as 15 an multiply it by 1 1/2 you get 22 1/2

Answer:

Scalene, isosceles and equilateral triangle are the types of triangles

Step-by-step explanation:

They  differ from each other based on their side-length. If all the three sides are different in length, then its scalene triangle. If any two sides are equal in length, then it is an isosceles triangle.

Answer:

Your question seems to be wrong, I'm the equation given there is no arithmetic sign between the constant and the value with the coefficient of X.

Step-by-step explanation:

Anyways, to solve this, you just input 1 as the value of X in the equation. When this is done, you solve normally.

The graph of y=3x is a straight line, it can be graphed by taking two points on the line.

What is a graph?

A diagram that depicts the relationship between two variables, usually measured along opposite axes in a pair, is used.

The equation we have is y=3x.

Since it is a linear equation, the graph will be a straight line.

A Line can be plotted with two points.

The two points on line y=3x are (0,0) and (1,3)

Because when x=0, y=0 and  when x=1, y=3.

We have to plot the two points on the graph and join the points and extend the line segment.

This way we can graph y=3x.

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Answer:

Step-by-step explanation:

You want to know the intercepts of g(x) = -2f(x), where the intercepts of f(x) are (0, 5), (2, 0) and (-4, 0).

The scale factor of -2 applied to f(x) multiplies each of the y-coordinates by -2.

  (x, y) ⇒ (x, -2y)

  (0, 5) ⇒ (0, -10) . . . . y-intercept

  (2, 0) ⇒ (2, 0) . . . . . x-intercept

  (-4, 0) ⇒ (-4, 0) . . . . x-intercept

The intercepts of g(x) are (0, -10), (2, 0) and (-4, 0).

The intercepts of g(x) = -2f(x) are (0, -10), (2, 0), and (-4, 0).

The coordinates in a graph indicate the location of a point with respect to the x-axis and y-axis.

The coordinates in a graph show the relationship between the information plotted on the given x-axis and y-axis.

We have,

f(x) has a y-intercept at 5 and x-intercepts at 2 and −4.

This means,

(0, 5), (2, 0), and (-4, 0)

Now,

g(x) = -2f(x).

This means,

Vertical scaling.

Multiply -2 with the y-coordinate.

(0, 5) = (0, -10)

(2, 0) = (2, 0)

(-4, 0) = (-4, 0)

Now,

The intercepts are (0, -10), (2, 0), and (-4, 0).

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Answer: The square of 2[tex]\sqrt{3[/tex] is 12

Step-by-step explanation:

squaring the 2 and [tex]\sqrt{3}[/tex] , then multiplying the results

[tex]2^{2}[/tex] =4

again [tex]\sqrt{3} ^{2}[/tex]= 3

therefore, 4x3 =12

Add 5 to 2 = 2 will make 7 = 7 a true statement and adding 5 to 2 ≠ 3 to make 7 ≠ 8 will make a true statement.

The number system is a way to represent or express numbers.

A decimal number is a very common number that we use frequently.

Any of the multiple sets of symbols and the guidelines for utilizing them to represent numbers are included in the Number System.

(1)

As per the given,

2 = 2

Add 5 on both sides of the above equation,

2 + 5 = 2 + 5

7 = 7 (True)

(2)

As per the given

2 ≠ 3

Add 5 on both sides of the above equation,

2 + 5 ≠ 3 + 5

7 ≠ 8 (True)

Hence "It is correct that if you add 5 to 2 = 2, you will get 7 = 7, and if you add 5 to 2 ≠ 3, you will get 7 ≠ 8".

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