What is nature of roots formula?

The value of k in the quadratic equation is k = 0 or k = 8

What is Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

where ( b² - 4ac ) is the discriminant

when ( b² - 4ac ) is positive, we get two real solutions

when discriminant is zero we get just one real solution (both answers are the same)

when discriminant is negative we get a pair of complex solutions

Given data ,

Let the quadratic equation be represented as A

Now , the value of A is

2x² + kx + k = 0   be equation (1)

And , the roots of the equation are equal

So , when the discriminant is zero we get just one real solution

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

( b² - 4ac ) = 0

Substituting the values in the equation , we get

k² - 4 ( 2 ) ( k ) = 0

On simplifying the equation , we get

k² - 8k = 0

Adding 8k on both sides of the equation , we get

k² = 8k

So , when k = 0 , the equation satisfies

And ,

k = 8

Therefore , the value of k are 0 and 8

Hence , the value of k in the equation is 0 or 8

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The last statement the ratio of smoothies size to mangos used for Kim is same as the ratio of smoothie size to mangos used for Angela is correct.

What is ratio?

A ratio in mathematics demonstrates how many times one number is present in another. For instance, if a bowl of fruit contains eight oranges and six lemons, the ratio of oranges to lemons is eight to six. The ratio of oranges to the total amount of fruit is 8:14, and the ratio of lemons to oranges is 6:8.

Given:

Kim and her friends watched the server making smoothies.

The given table shows the number of mangos that were used for each of the different size of smoothies that the friends ordered.

From the given statements the last statement 'the ratio of smoothies size to mangos used for Kim is same as the ratio of smoothie size to mangos used for Angela.'

Because,

The ratio of smoothies size to mangos used for Kim is, 21/3 = 7 and

The ratio of smoothie size to mangos used for Angela is 28/4 = 7.

Both the ratios are same.

Hence, the last statement the ratio of smoothies size to mangos used for Kim is same as the ratio of smoothie size to mangos used for Angela is correct.

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Answer:

[tex]\frac{2}{15} (3+\sqrt[3]{3})[/tex]

or

[tex]\frac{6+2\sqrt[3]{3} }{15}[/tex]

You can rationalize the denominator to simplify the answer so many different ways.

Step-by-step explanation:

Simplify the radicals.

(x-1), (x-2) and (x+3) are the three factors of the above cubic polynomial. A degree three polynomial might be referred to as a cubic polynomial.

A degree three polynomial might be referred to as a cubic polynomial. In other words, the number three will be the largest exponent of the variable in a cubic polynomial. A cubic polynomial would therefore have the generic form ax3+bx2+cx+d, where a ≠0.

here

Let p(x)=[tex]x^{3}[/tex]−7x+6

From figure, we have p(1)=0

Thus, (x−1) is a factor of p(x)

The other factor is [tex]x^{2}[/tex]+x−6

=(x+3)(x−2)

Therefore, [tex]x^{3}[/tex]−7x+6 =(x−1)(x+3)(x−2)

(x-1), (x-2) and (x+3) are the three factors of the above cubic polynomial.

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Answer:

Step-by-step explanation:

A discontinuous function can be solved by analyzing the behavior of the function and its points of discontinuity. This can be done by finding the limits of the function at each point and then using the appropriate algebraic techniques to solve the equation.

Discontinuous functions are those that have gaps or sharp discontinuities in their graphs. In order to solve a discontinuous function, you must start by analyzing the behavior of the function and its points of discontinuity. This can be done by finding the limits of the function at each point and then using the appropriate algebraic techniques to solve the equation. This can involve factoring, manipulating equations, and other techniques to find the solution. Once the equation is identified, it can then be graphed to see the behavior of the function between the points of discontinuity. If necessary, the equation can then be further manipulated to satisfy the conditions of the discontinuous points. By approaching the problem in this manner, a discontinuous function can be effectively solved.

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Answer:

2a. Scale factor is 2

2b.  Scale factor is 3

Step-by-step explanation:

2a.

We see that the smaller figure to get from F' to G', it is 2 squares, and we see the larger figure to get from F to G, it is four squares, so the scale factor is 2.

2b.

We see that to get from A' to C', it is 2 squares, and we see the larger figure to get From A to C, it is 6 squares, so the scale factor is 3

An example of continuous data is the temperature of a room over time. It can change gradually, or even rapidly, but the data is always changing.

Continuous data is data that can take on any value within a given range. It can be measured on an infinite scale and can have an infinite number of values. An example of continuous data is the temperature of a room over time. This data can change gradually or rapidly, but it is always changing. Continuous data can be measured directly, such as measuring the temperature of a room with a thermometer, or indirectly, such as measuring the temperature of a room through observation. Continuous data can also be used to describe physical or psychological phenomena, such as measuring a person’s heart rate over time or measuring the concentration of a chemical in a solution. Continuous data is often used in scientific research, as it allows researchers to observe trends and correlations over time.

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The value of the polynomial 5x- 4x²+3 at x=-1 is -6 by using the substitution method.

Given that,

The polynomial is 5x- 4x²+3

We have to find the value of the polynomial when x=-1.

We know that,

A polynomial is an algebraic expression whose exponents are all whole numbers. There must be non-negative integer variable exponents in all polynomials.

Take the polynomial P(x) = 5x- 4x²+3

P(x) = 5x- 4x²+3

So, we take x=−1,

P(−1)=5×(−1)−4×(−1)²+3

P(−1)=−5−4+3

P(−1)=−6

Therefore, The value of the polynomial 5x- 4x²+3 at x=-1 is -6 by using the substitution method.

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The formula can be used to find the missing number in an array of 1 100 is N * (N + 1) / 2.

The size of the array is N – 1. So the sum of n elements, that is the sum of numbers from 1 to N can be calculated by using the formula N * (N + 1) / 2. Now find the summation of all elements in the array and subtract it from the summation of first N natural numbers, the value obtained will be the value of the missing element.

The formula can be used to find the missing number in an array of 1 100 is N * (N + 1) / 2.

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The value of the percent increase in the value of W is 50%.

How can I determine the ratio between two quantities?

When the ratio of each of the two quantities' values is equal, the two quantities are said to be proportionate. It comes in two varieties: direct and indirect. The ratio is more than one for direct relations whereas it is less than one for indirect relations.

The relationship provided in the issue can be expressed as,

W ∝ X

X now has a new value when it is increased by 50%:

X + 50% × X

⇒ X + 50/100 × X = 3X/2

Assume that X and W now have values of X1 and W1, respectively.

Consequently, they are related in the following ways:

W ∝ X

⇒ 3/2W ∝ 3/2X

⇒ 3/2W ∝ X₁

W's value is increased by 3/2W - W, which equals 1/2W.

The percentage growth

⇒ (1/2W)/W × 100%  = 50%

Hence, the percent increase is obtained as 50%.

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The relationship between the volume of the rectangular pyramid and the volume of the rectangular prism with the same base and height is given by : Volume of the rectangular pyramid = (1/3) of the volume of the rectangular prism.

As given in the question,

Formula need to use for the calculation of the volume of the rectangular pyramid and prism is  :

Volume of the rectangular pyramid

= (1/3) × ( rectangular base area) × height

Volume of the prism = ( rectangular base area ) × height

If the base and height of the rectangular pyramid and rectangular prism is same then,

Volume of the rectangular pyramid

= ( one - third ) × Volume of the rectangular prism

Therefore, the relationship representing the volume of the rectangular pyramid is and prism with same base and height as volume of pyramid is one third the volume of the prism.

The above question is incomplete , the complete question is:

What is the relationship between the volume of rectangular prism and pyramid which have the same height and base?

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The number of oranges needed to make 3quarts of oranges juice is 20oranges

A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 4 girls you could write the ratio as: 1 : 4(for every one boy there are 3 girls). This also means that for 2 boys there will be 6 girls.

Similarly 5 oranges is needed to make 3 cups of juice. it can also be put in like 5:3

to make 3quarts,

2cups = 1pint

2pints = 1quart

therefore 4 cups = 1quart

3 quart = 3×4=12cups

if 3 cups is from 5 oranges

then 1 cup = 5/3

12cups = 5/3× 12= 20 oranges

Therefore 20oranges will be needed to make 12 cups of juice

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The relationship between the volume of a sphere and the volume of a cylinder is said to be that the volume of any sphere is 2/3rd of the volume of any with equal radius and height which is equal to the diameter.

Assuming a cylinder with radius r, height h.

We are aware that the cylinder-volume formula is,

V₁ = π * r² * h

Consider a sphere with radius r₁.

We know that the formula for the volume of sphere is,

V₂ = 4/3 * π * r₁³

If the sphere and a cylinder have same radii and height of cylinder equal to the diameter then, r₁ = r and h = 2r

V₂ = 4/3 * π * r³

And V₁ = π * r² * (2r)

V₁ = 2 * π * r³

V₁ = 2 * 3/4 * V₂

V₁ = 3/2 * V₂

Volume of cylinder = (3 / 2) *  Volume of sphere

Therefore, the Volume of a sphere = 2/3 * Volume of a cylinder

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Answer:

Below

Step-by-step explanation:

Domain is the x values a graph can have

   this one goes from 7 to 11   inclusive ( solid dots at each end)

7<= x <= 11  

To find the interest rate, we can use the formula for simple interest:

I = P * r * t

Where I is the interest, P is the principal (the amount of money in the account), r is the interest rate, and t is the time (in years).

We know that I = $480, P = $2,000, and t = 2 years.

Plugging these values into the formula, we get:

$480 = $2,000 * r * 2

Dividing both sides by $2,000 * 2, we get:

r = $480 / $2,000 * 2

r = 0.12

So, the interest rate is 0.12, or 12%.

If A and B are independent events with given probabilities P(A) and P(B), then the probability of A intersecting B (P(A ∩ B)) is equal to the product of their individual probabilities. For example, if P(A) = 0.05 and P(B) = 0.65, then P(A ∩ B) = 0.0325.

The probability of two events A and B occurring together, P(A ∩ B), can be calculated by multiplying the individual probabilities of each event, P(A) and P(B). In this case,

P(A) = 0.4 and P(B) = 0.6, so P(A ∩ B) = 0.4 * 0.6 = 0.24

.The probability of event A occurring given that event B has already occurred, P(A | B), is equal to the probability of both events occurring together, P(A ∩ B), divided by the probability of event B occurring, P(B). In this case,

P(A ∩ B) = 0.24 and P(B) = 0.65, so P(A | B) = 0.24 / 0.65 = 0.0375

which can be rounded to 0.0325.These probabilities can be used to estimate the likelihood of certain events occurring together or given certain conditions. In this case, the probability of events A and B occurring together is 0.24 and the probability of event A occurring given that event B has already occurred is 0.0325.

A. P(A ∩ B) = P(A) * P(B)

= 0.4 * 0.6

= 0.24

B. P(A | B) = P(A ∩ B) / P(B)

= 0.24 / 0.65

= 0.0375

= 0.0325

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We can reduce the margin of error in an interval estimate of p by doing any of the following except using a planning value p* closer to 0.5.

The breadth of the confidence interval is affected by the following features:

Correct Question:

We can reduce the margin of error in an interval estimate of p by doing any of the following except ____.

increasing the sample size

using a planning value p* closer to 0.5

increasing the level of significance

reducing the confidence coefficient

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Answer: C

Step-by-step explanation:

(-3x + 9i) (x + 3i)

-3x^2 - 9ix + 9ix - 27

-3x^2 - 27

So, the answer is C

The Side-Side-Side (SSS) criterion states that two triangles are similar if "sides of one triangle in two triangles are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and so the two triangles are comparable."

Look at the triangles in the picture below. We can observe that the triangles' three sets of side pairs are all congruent. "SSS" or "side-side-side" are other names for this. Two triangles are congruent if they have three pairs of congruent sides, according to the SSS requirements for triangle congruence.

Congruence Rule Theorem for SSS

Two triangles are said to be congruent if the corresponding three sides of one triangle match the corresponding three sides of the other triangle exactly.

We shall now demonstrate the aforementioned theorem.

Given: [tex]\triangle A B C[/tex] and [tex]\triangle P Q R[/tex] such that AB=PQ, BC=QR and AC=PR.

To prove: [tex]\triangle A B C \cong \triangle PQR[/tex]

Construction: Let BC be the longest side of [tex]\triangle A B C[/tex] and so QR is the longest side of [tex]\triangle P Q R.[/tex]

Draw PS so that [tex]\angle R Q S=\angle C B A[/tex] and [tex]\angle Q R S=\angle B C A[/tex].

Join SQ and SR.

In [tex]\triangle ABC[/tex] and [tex]\triangle SQR[/tex],

BC=QR Given

[tex]\angle C B A=\angle R Q S[/tex] By construction

[tex]\angle B C A=\angle Q R S[/tex] By construction

[tex]\triangle A B C \cong \triangle SQR[/tex] By ASA congruence

[tex]\angle A=\angle S[/tex] By CPCTC

AB=SQ By CPCTC

AC=SR

Now, AB=PQ and AB=SQ => PQ=SQ

Similarly, AC=PR and AC=SR => PR=SR

Hence, [tex]\triangle ABC \cong \triangle PQR,[/tex]

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Yes, the equation has real roots. By completing the square, we can make the equation in the form of ax^2+bx+c=0. Then, we can use the quadratic formula to solve for x. The discriminant of this equation is b^2-4ac = 10-60 = -50, which is negative, so the equation has two real roots.

Yes, the equation 3x^2+2√5x has real roots. By expanding the equation, we get 3x^2+2x√5=0. We can use the quadratic formula to solve for x. First, we need to make the equation in the form of ax^2+bx+c=0. To do this, we complete the square. We divide both sides of the equation by 3, giving us x^2+2/3*√5x=0. Then, we add (2/3*√5/2)^2 to both sides, giving us x^2+2/3*√5x+(2/3*√5/2)^2= (2/3*√5/2)^2. Simplifying, we get x^2+2/3*√5x+5/12=0. This is now in the form of ax^2+bx+c=0, with a=1, b=2/3*√5, and c=5/12.

Now, we can use the quadratic formula to solve for x

x = [-2/3*√5 + √(-50/3)]/2

x = [-2/3*√5 - √(-50/3)]/2

Therefore, the two real roots of the equation are

x = [-2/3*√5 + √(-50/3)]/2 and

x = [-2/3*√5 - √(-50/3)]/2.

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Whole Numbers are the numbers that include natural numbers and zero. Not a fraction or decimal. {0, 2, 3, 4, 5 6, 7, 8, 9, 10, 11 …}.

Whole Numbers

Whole numbers are a set of numbers including all natural numbers and 0. They are a part of real numbers that do not include fractions, decimals, or negative numbers. Counting numbers are also considered as whole numbers.

Addition , Subtraction , Division and multiplication are also possible with the whole numbers as they are also numbers . Addition and Multiplication are closed which means the sum and multiplication of two numbers is also a whole number whereas Subtraction and Division are unclosed.

We know that natural numbers refer to a set of counting numbers starting from 1, 2, 3, 4 and so on. In simple words, whole numbers are a set of numbers without fractions, decimals, or even negative integers.

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A dilation is a type of transformation that enlarges or reduces a figure.It is also called pre image.To create new figure is called image.For this purpose we can use scale factor ,r how much bigger or smaller the dilation image will be compared to pre-image.

It is a transformation which is used to resize the image.It is used to make objects smaller or greater.But there is no difference in the size of the shape.The pre -image and image is not congruent.

For this purposes we use scale factor.

When you dilates a image the scale factor find that how much image will be smaller or larger.

.If the scale factor is between o0 and 1 .Then image is smaller than preimage.Dilation with a scale factor between 0 and 1 i called transformation.

.if the scale factor is greater than 1.Then image will be larger than the preimage.

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Multiplicity is a mathematical concept related to the roots of a polynomial equation.

It is defined as the number of times a given root is repeated in the polynomial equation.

The formula for calculating the multiplicity of a root is :

Multiplicity = n-(degree of the polynomial equation)

where n is the order of the root and the degree of the polynomial equation is the highest exponent in the equation.

For example, if the polynomial equation is 3x^2 + 4x + 2 = 0 and the root is 2, then the multiplicity of the root is calculated as follows:

Multiplicity = n - (degree of the polynomial equation)

Multiplicity = 2 - 2

Multiplicity = 0

This means that the root 2 is not repeated in the polynomial equation and therefore has a multiplicity of 0.

In another example, if the polynomial equation is 4x^3 + x^2 + 6x + 12 = 0 and the root is 2, then the multiplicity of the root is calculated as follows:

Multiplicity = n - (degree of the polynomial equation)

Multiplicity = 2 - 3

Multiplicity = -1

This means that the root 2 is repeated in the polynomial equation and therefore has a multiplicity of -1.

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f(x) = 2[tex]x^{3}[/tex] + 4[tex]x^{2}[/tex] - 22x -24 will be the equation.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign. For example, 3x – 5 = 16 is an equation. Solving this equation, we get the value of the variable x as x = 7.

The equation will be 2(x-3)(x+4)(x+1) since the function f(x) has roots of 3, -4, and -1 and a leading coefficient of 2.

so 2(x-3)(x+4)(x+1) will be the equation

That is 2(x-3)([tex]x^{2}[/tex] +5x + 4)

2([tex]x^{3}[/tex] + 2[tex]x^{2}[/tex] - 11x -12)

f(x) =2[tex]x^{3}[/tex] + 4[tex]x^{2}[/tex] - 22x -24

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The 3 consecutive odd numbers whose sum will be 51 are  16, 17, 18.

What are consecutive numbers ?

Numbers that follow each other continuously in the order from smallest to largest are called consecutive numbers. For example: 1, 2, 3, 4, 5, 6, and so on are consecutive numbers.

Since, we know that any odd number is always of the form (2k +1). We express an odd number in this form because we know that an odd number is not divisible by 2.

Let the numbers be x, x+1 and x+2.

Then the sum is : x + x+1+ x+2 = 3x + 3

3x + 3 = 51 (given)

3x = 51 – 3 = 48

x = 48 ÷ 3

x = 16

∴

Hence , the 3 consecutive odd numbers whose sum will be 51 are  16, 17, 18.

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The heights of the two legs (a and b) of a right triangle are equal. Ha = b and hb = an as a result. The height of the hypotenuse is hc. The intersection of a triangle's three altitudes occurs at the orthocenter H, which for a right triangle is located on the vertex C of the right angle.

A line segment passing through a triangle's vertex and running perpendicular to the line containing the base is the triangle's height in geometry.

The extended base of the altitude is the name given to this line that contains the opposing side.

The foot of the altitude is the point at which the extended base and the height converge.

Each leg (a and b) of a right triangle has the same altitude as its opposite leg.

As a result, ha = b and hb = a.

The hypotenuse rises to an altitude of hc.

The orthocenter H, which for a right triangle lies on the vertex C of the right angle, is where the three altitudes of a triangle intersect.

Therefore, the heights of the two legs (a and b) of a right triangle are equal. Ha = b and hb = an as a result. The height of the hypotenuse is hc. The intersection of a triangle's three altitudes occurs at the orthocenter H, which for a right triangle is located on the vertex C of the right angle.

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The four parts of a line graph is Tittle, Legend, Data, Y-axis, X-axis and Line.

A line graph—also called a line plot or a line chart—is a graph that makes use of lines to connect individual data points. A line graph presentations quantitative values over a distinctive time of language. In finance, line graphs are normally used to depict the ancient rate action of an asset or security. Line graphs may be compared with different visualizations of facts which includes bar charts, pie charts, and (in buying and selling) candlestick charts, amongst others.

Line graphs might also vary relying non-obligatory capabilities or formatting. the very best-excellent, easiest to understand line graphs have the subsequent characteristics:

Line graphs may additionally have a name above the graph to succinctly provide an explanation for what the graph is depicting except we provide a user with written context, the user will regularly depend on the title to better understand what information is being pulled in. The title can also mainly name out a time frame or limits to the information.

The legend explains what every based variable is and the way to distinguish extraordinary units of data. In the instance above, every dependent variable is marked with its own coloration. The field that explains what each coloration method is the legend.

Each object of facts on a line graph is a reference to a distinctive supply that ties the structured variable to an unbiased variable. that is the records on our graph; it is the item that creates the dots that get linked to form the lines to our chart. There may be a couple of units of statistics blended into a single graph. To make sure statistics is blanketed and correct, companies might also have precise information integrity analyst or similar positions to screen database hobby.

The x-axis is the set of records that runs along the horizontal, flat component at the bottom of the road graph. In most line graphs, the x-axis could be related to time, whether it's far the distinctive months in a year or the wide variety of weeks which have exceeded considering that a product launch.

The y-axis is the set of data that runs alongside the vertical, left-aspect of the graph. some iterations of line graphs have this set of data on the proper. In any case, these numbers count number the items being measured. The graph might also begin at 0, even though there are instances wherein it makes more experience to begin at a higher variety.

Finally, the line. the road connects all statistics points within a single structured variable. This line's motion shows the growth and reduce of information throughout time. it may also easily be as compared towards different strains so long as all facts units are being measured over similar periods of time. Although overly simplified, this line can speak to management what actions need to be taken to enhance operations or strategic planning.

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You must have held Social Security-covered employment and have a medical condition that satisfies Social Security's rigid definition of disability in order to be eligible for SSS disability benefits.

People who are unable to work for a year or more due to a handicap typically receive monthly payments. Benefits often persist until you can resume regular employment. Additionally, there are a number of unique regulations, referred to as work incentives, that offer ongoing benefits and health insurance to support your move back to employment.

When you reach full retirement age while receiving SSDI payments, your disability benefits will automatically transition to retirement benefits, but the amount will not change.

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