igcse math (quadratic functions)
given the equation: x^2 + 5x - 14
use the discriminant to decide if there are two real and different roots, two equal roots or no real roots

To find the area inside the larger loop and outside the smaller loop of the limaçon r = 1 2 + cos(θ), we need to first visualize the graph of the limaçon.

The equation r = 1 2 + cos(θ) represents a curve that resembles a snail shell or a heart shape with a loop inside another loop. To find the area inside the larger loop and outside the smaller loop, we need to set up the integral using the polar coordinates.

The formula for the area enclosed by a polar curve is given by:
A = 1/2 ∫(θ2-θ1) (r2-r1)² dθ, where r2 is the outer curve, r1 is the inner curve, and θ1 and θ2 are the angles where the curves intersect.

In this case, the inner curve is r = 1 - cos(θ), and the outer curve is r = 1/2 + cos(θ). The curves intersect when cos(θ) = 1/2 or θ = π/3 and θ = 5π/3.
So, we need to split the integral into two parts, one for θ = π/3 to θ = 5π/3, and another for θ = 5π/3 to θ = π/3.

This is because the outer curve becomes the inner curve and vice versa when we cross the angle θ = 5π/3. For the area inside the larger loop and outside the smaller loop, we need to subtract the area enclosed by the inner curve from the area enclosed by the outer curve.

Using the formula above and plugging in the values, we get:
A = 1/2 ∫(5π/3-π/3) [(1/2 + cos(θ))² - (1-cos(θ))²] dθ

Simplifying this integral, we get:
A = 1/2 ∫(5π/3-π/3) [5/4 + 2cos(θ)] dθ
A = 5/8 [θ + sin(θ)](5π/3-π/3)
A = 5/8 [4π/3 + sin(4π/3) - (π/3 + sin(π/3))]
A = 5/8 [4π/3 - √3/2]
A = 5π/6 - (5/16)√3
Therefore, the area inside the larger loop and outside the smaller loop of the limaçon r = 1 2 + cos(θ) is 5π/6 - (5/16)√3.

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To find the area inside the larger loop and outside the smaller loop of the limaçon r = 1 + 2cos(θ), use the formula for finding the area enclosed by a polar curve.

The general formula is A = (1/2)∫(r^2)dθ, where r is the equation of the curve. In this case, we need to find the area between two curves: the larger loop given by r = 1 + 2cos(θ) and the smaller loop given by r = 1. To find the limits of integration, we need to find the values of θ where the two curves intersect. After finding the values of θ where the curves intersect, we can integrate the difference between the two equations r = 1 + 2cos(θ) and r = 1 with respect to θ.

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Joaquin will need 1 and two fifths cups to make his famous chocolate chip cookies for his friend's birthday party

To solve this problem we will use a rule of three with the problem information:

5 people-------- 1/4 cup of butter

28 people -------- x

Applying the rule of three we get:

x = ( 28 people * 1/4 cup of butter) / 5 people

x = 1,4 cup of butter

x = 1 + 2/5 cup of butter = 1 and two fifths cups

It describes the proportionality of 3 known data and an unknown data. When you have more than 3 known facts that are involved in the proportionality, it is known as a compound rule. The rule of three is also known as a direct proportions.

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Step-by-step explanation:

48 oranges in 5 minutes means she picked 48 oranges in 300 seconds.

Divide 48 by 300 to get what she would pick in one second.

This equals 0.16

Multiply 0.16 by 120 seconds [how many seconds are in two minutes].

This equals 19.2 oranges in two minutes. As she cannot pick 0.2 of an orange, she could pick 19.

A finite group is a group that has a finite number of elements. Now, let us define the center of a group. The center of a group, denoted by Z(G), is the set of all elements in G that commute with every element in G.

Now, we need to prove that if g is a finite group, the index of Z(g) cannot be prime. We can prove this using contradiction. Suppose the index of Z(g) is prime. Let this prime be denoted by p. This means that the number of distinct left cosets of Z(g) in g is p. Therefore, we can write:

|g/Z(g)| = p

where |g/Z(g)| represents the number of distinct left cosets of Z(g) in g.

Now, we can use the fact that the number of left cosets of a subgroup in a group is equal to the index of that subgroup in the group. Therefore, we can rewrite the above equation as:

|g|/|Z(g)| = p

Multiplying both sides by |Z(g)|, we get:

|g| = p|Z(g)|

Since p is a prime number, it can only be divided by 1 and itself. Therefore, the only possible divisors of p|Z(g)| are 1, p, and |Z(g)|.

Now, since |g| is finite, we know that |Z(g)| cannot be infinite. Therefore, the only possible values for |Z(g)| are positive integers that divide |g|. However, since p is a prime number, |Z(g)| cannot be equal to p. This means that the only possible values for |Z(g)| are 1 and |g|.

If |Z(g)| = 1, this means that Z(g) only contains the identity element. Therefore, g does not have any non-identity elements that commute with every other element in g. This is not possible since every group has at least one element that commutes with every other element in the group - the identity element.

If |Z(g)| = |g|, this means that every element in g commutes with every other element in g. This implies that g is an abelian group. However, this contradicts the fact that g is a finite group that is not abelian.

Therefore, we have reached a contradiction in both cases. This means that our assumption that the index of Z(g) is prime is false. Therefore, if g is a finite group, the index of Z(g) cannot be prime.

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Answer:

x = 12

Explanation:

Given the expression

17-5(2x-9)=-(-6x+10)+4

​Open the parenthesis

17 - 5(2x) - 5(-9) = -6x + 10 + 4

17 - 10x + 45 = -6x + 14

-10x + 62 = -6x + 14

Collect the like terms

-10x + 6x = 14 - 62

-4x = -48

Divide both sides by -4

-4x/-4 = -48/-4

x = 48/4

x = 12

Hence the value of x that makes the equation true is 12

If the discriminant is 0, then you'll have exactly one root. This is a double root since the root repeats itself.

Example: y = (x-5)^2 = x^2-10x+25 has a double root at x = 5

For this following function by evaluating f(5)-f(4) we get -19.

What is evaluate?

Finding the value of an algebraic expression when a given number is used to replace a variable is known as evaluating the expression. When evaluating an expression, we change the variable in the expression to the given number and then use the order of operations to simplify the expression.

Sol- f(5)-f(4)

f(5)-f(4)= (-2.5^2-5+3)-(-2.4^2-4+3)

Calculate the power

(-2×25-5+3)-(-2×4^2-4+3)

Calculate the product or quotient

(-50-5+3)-(-2×16-4+3)

Calculate the sum

-52-(-32-4+3)

=-52-(-33)

determine the sign

=-52+33

By calculating the sum of difference we are get

-19.

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(a) 95% confidence interval for the difference between female and male times is (11.954, 255.591).

(b) The assumptions and conditions for the two-sample t-test are met, so we can use the results of the test and confidence interval.

a) To construct a 95% confidence interval for the difference between female and male times, we can use a two-sample t-test. Let's denote the mean time for female swimmers as μf and the mean time for male swimmers as μm. We want to test the null hypothesis that there is no difference between the two means (i.e., μf - μm = 0) against the alternative hypothesis that there is a difference (i.e., μf - μm ≠ 0).

The formula for the two-sample t-test is:

t = (Xf - Xm - 0) / [sqrt((s^2f / nf) + (s^2m / nm))]

where Xf and Xm are the sample means for female and male swimmers, sf and sm are the sample standard deviations for female and male swimmers, and nf and nm are the sample sizes for female and male swimmers, respectively.

Using the data from the plots, we get:

Xf = 917.5, sf = 348.0137, nf = 15

Xm = 783.7273, sm = 276.0625, nm = 28

Plugging in these values, we get:

t = (917.5 - 783.7273 - 0) / [sqrt((348.0137^2 / 15) + (276.0625^2 / 28))] = 2.4895

Using a t-distribution with (15+28-2) = 41 degrees of freedom and a 95% confidence level, we can look up the critical t-value from a t-table or use a calculator. The critical t-value is approximately 2.021.

The confidence interval for the difference between female and male times is:

(917.5 - 783.7273) ± (2.021)(sqrt((348.0137^2 / 15) + (276.0625^2 / 28)))

= 133.7727 ± 121.8187

= (11.954, 255.591)

Therefore, we can be 95% confident that the true difference between female and male times is between 11.954 and 255.591 minutes.

b) Assumptions and conditions for the two-sample t-test:

Independence, We assume that the observations for each group are independent of each other.

Normality, We assume that the populations from which the samples were drawn are approximately normally distributed. Since the sample sizes are relatively large (15 and 28), we can rely on the central limit theorem to assume normality.

Equal variances, We assume that the population variances for the female and male swimmers are equal. We can test this assumption using the F-test for equality of variances. The test statistic is,

F = s^2f / s^2m

where s^2f and s^2m are the sample variances for female and male swimmers, respectively. If the p-value for the F-test is less than 0.05, we reject the null hypothesis of equal variances. If not, we can assume equal variances. In this case, the F-test yields a p-value of 0.402, so we can assume equal variances.

Sample size, The sample sizes are both greater than 30, so we can assume that the t-distribution is approximately normal.

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Answer:

8

Step-by-step explanation:

The there smallest consecutive odd numbers are 1,3 and 5

Therefore the smallest possible perimeter of such triangle = 8

Answer:

Step-by-step explanation:

x / 12 = 3 / 8

x = 3 / 8 x 12

x = 3 / 94

x = 32

32 / 12 = 3 / 8

Answer:

\(x=-2\\x=2\)

(You only need to give one solution)

Step-by-step explanation:

We have the following equation

\((x^2+1)^2-5x^2-5=0\)

First, we need to foil out the parenthesis

\(x^4+2x^2+1-5x^2-5=0\)

Now we can combine the like terms

\(x^4-3x^2-4=0\)

Now, we need to factor this equation.

To factor this, we need to find a set of numbers that add together to get -3 and multiply to give us -4.

The pair of numbers that would do this would be 1 and -4.

This means that our factored form would be

\((x^2-4)(x^2+1)=0\)

As the first binomial is a difference of squares, it can be factored futher into

\((x^2+1)(x+2)(x-2)=0\)

Now, we can get our solutions.

The first binomial will produce two complex (Not real) solutions.

\(x+2=0\\\\x=-2\)

\(x-2=0\\\\x=2\)

So our solutions to this equation are

\(x=-2\\x=2\)

Considering that each input is related to only one output, the correct option regarding whether the relation is a function is:

A. yes.

A relation represents a function when each value of the input is mapped to only one value of the output.

For this problem, we have that:

There are no repeated inputs, hence the relation is a function and option A is correct.

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Answer:

11x2+6x

Step-by-step explanation:

Answer: its either 21x^2 +2x or 11x^2-6x Im not sure which one is solved correctly :/

Step-by-step explanation:

8x^2+4x-6x^2+2+7x^2-2x

21x^2 +2x

or

(8x^2+4x-6x2)2+7x^2-2x

-12x^2+8x+16x^2+7x^2-2x

11x^2-6x

Answer: 17:15

Step-by-step explanation:

From the question, we are informed that the ticket booth at a school play sells 170 adult tickets and 150 student tickets.

The ratio of adults to students will be:

= 170/150

= 17:15

The ratio is therefore 17:15.

Answer:

$39.36

Step-by-step explanation:

Since its 20% off that means we still have a remaining 80% so first multiply 0.8 to the 82

82x0.8= 65.6

Since there is a additional 40% off multiply by 0.4 to get the discount amount

65.6x0.4= 26.24

Subtract the discount from the price

65.6-26.24= 39.36

Answer:

okay so the way your going to set this up is backwards because you already know your hypotenuse. 65^2 = 25^2 + b^2. So its 24 feet

Step-by-step explanation:

Answer:

-1/5

Step-by-step explanation:

m=(y2-y1)/(x2-x1)

m=(-1-0)/(5-0)

m=-1/5

The value of ∠WXY = 20.

What is Exterior angle theorem?

The exterior angle theorem describes the connection between the two remote angles in a triangle and the external angle created by an extended side outside the triangle.

Given: Measure of angle YWX = (3x + 17) °

Measure of angle WXY = (3x + 2) °

Measure of angle XYZ = (10x − 5) °

Therefore, m∠XYZ = m∠YWX + m∠WXY (exterior angle theorem)

⇒ (10x − 5) ° = (3x + 17) ° + (3x + 2) °

Solve for x,

⇒ 10x - 5 = 3x + 17 + 3x + 2

⇒ 10x - 6x = 17 + 7

⇒ 4x = 24

⇒ x = 6

∴ ∠WXY = (3x + 2) = 18 + 2 = 20

Hence, value of ∠WXY = 20.

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Answer:

35%

Step-by-step explanation:

There are 20 sqaures, 7 of them are not shaded, 7/20=35%

The empirical probability of drawing the cards will be 6 / 55.

The ratio of the number of outcomes in which a defined event occurs to the total number of trials, not in a theoretical sample space but in a real experiment, is the empirical probability, relative frequency, or experimental probability of an event.

Given that a card is drawn one at a time from a well-shuffled deck of 52 cards. In 13 repetitions of this experiment, 1 king is drawn.

The number of kings in a well-shuffled deck consists of 52 cards which is 4.

The number of ways of drawing consists of 4 kings in 13 repetitions which is ¹³C₄.

In 13 repetitions, 2 kings are drawn by ¹³C₂ ways,

The empirical probability will be calculated as,

P(E) = ¹³C₂ / ¹³C₄

P(E) = [ (13!) / (13-2)! ] ÷ [ (13!) / ( 13-4)!(4!) ]

P(E) = ( 4 x 3 ) / ( 11 x 10)

P(E) = 6 / 55

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If 300 people were sampled instead of 200, and the confidence level remained the same, it would produce a more accurate result.

Sampling means selecting the group that you will actually collect data from in your research. For example, if you are researching the opinions of students in your university, you could survey a sample of 100 students. In statistics, sampling allows you to test a hypothesis about the characteristics of a population.

This is because a larger sample size allows for a better representation of the population, providing a more accurate result. Additionally, with a larger sample size, the confidence interval of the sample would be narrower, indicating a higher level of confidence in the accuracy of the result.

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Two congruent triangles with the same orientation aligned with one above the other. The transformation is -

Option A: reflection over a horizontal line

What is reflection?

A reflection is referred to as a flip in geometry. A reflection is the shape's mirror image. A line, called the line of reflection, will allow an image to reflect through it. Every point in a figure is said to reflect the other figure when they are all equally spaced apart from one another.

In the figure it is given that -

Both the triangles are equal and congruent to each other.

There is no change in the shape or size of both the images only the place of the image is changed.

One triangle is placed above the another.

This is done over a horizontal plane.

Therefore, the transformation is reflection over horizontal line.

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Answer: $11.25h + $1000

Don't really have enough information to create a full equation please answer with what the equation is trying to solve and I will try to help further.

Given:

\(\Delta DHE\sim \Delta DGF\)

To find:

The value of x.

Solution:

We know that, corresponding sides of similar triangles are proportional.

Since,\(\Delta DHE\sim \Delta DGF\), therefore

\(\dfrac{HE}{GF}=\dfrac{DE}{DF}\)

On substituting the values from the figure, we get

\(\dfrac{8}{12}=\dfrac{x}{x+2}\)

\(\dfrac{2}{3}=\dfrac{x}{x+2}\)

On cross multiplication, we get

\(2(x+2)=3x\)

\(2x+4=3x\)

\(4=3x-2x\)

\(4=x\)

Therefore, the value of x is 4.

Answer: i believe its yes

Step-by-step explanation:

Answer:

B. Yes

Step-by-step explanation:

Answer:

The sum of 25 and 16 is 41.

Step-by-step explanation:

The sum of two numbers, 25 and 16, you can break apart the addends and add them separately to simplify the process. Here's how you can do it:

Break apart the numbers into their place values: For 25, you have 20 and 5, and for 16, you have 10 and 6. This step helps you work with the place values individually.

Add the tens place: In this case, you have 20 (from 25) and 10 (from 16). Adding them gives you 30.

Add the ones place: Now you add the ones place, which is 5 (from 25) and 6 (from 16). Adding them gives you 11.

Combine the sum of the tens place and the sum of the ones place: Take the sum of 30 (from step 2) and 11 (from step 3). Adding them together gives you 41.

So, the sun is 41.

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The sampling Distribution of the sample proportion of people who answer yes to the question is normal with mean p = 0.6 and standard deviation σp = 0.049.

Suppose 60% of American adults believe Martha Stewart is guilty of obstruction of justice and fraud related to insider trading. We will take a random sample of 100 American adults and ask them the question. Then the sampling distribution of the sample proportion of people who answer yes to the question is binomial.

The binomial distribution is appropriate since there are two possible outcomes: either a person believes that Martha Stewart is guilty of obstruction of justice and fraud related to insider trading or does not believe so. The sample size is n = 100.

The probability of a person believing that Martha Stewart is guilty is p = 0.6 and the probability of a person not believing is q = 1 - p = 0.4.

The mean of the sample proportion of people who believe Martha Stewart is guilty of obstruction of justice and fraud related to insider trading is given by:μp = p = 0.6

The variance of the sample proportion of people who believe Martha Stewart is guilty of obstruction of justice and fraud related to insider trading is given by:σ²p = pq/n = (0.6)(0.4)/100 = 0.0024

The standard deviation of the sample proportion of people who believe Martha Stewart is guilty of obstruction of justice and fraud related to insider trading is given by:σp = sqrt(σ²p) = sqrt(0.0024) = 0.049

This standard deviation measures how much the sample proportion is likely to vary from one sample to another. The sampling distribution of the sample proportion of people who answer yes to the question is approximately normal by the Central Limit Theorem (CLT) if the sample size is sufficiently large, which is the case here since np and nq are both greater than or equal to 10: np = (100)(0.6) = 60 and nq = (100)(0.4) = 40.

Therefore, the sampling distribution of the sample proportion of people who answer yes to the question is normal with mean p = 0.6 and standard deviation σp = 0.049.

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a. To find the probability that exactly four of the contracts experience cost overruns, we use the binomial probability formula:

P(X = 4) = (20 choose 4) * 0.2^4 * (0.8\()^16\)

where "X = the number of contracts that experience cost overruns". Using a calculator, we get:

P(X = 4) ≈ 0.2835

b. To find the probability that fewer than three of the contracts experience cost overruns, we need to find the probability that 0, 1, or 2 contracts experience cost overruns. We can use the binomial probability formula for each of these values and add the probabilities together:

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

= (20 choose 0) * 0.2^0 * (0.8)^20 + (20 choose 1) * 0.\(2^1\) * (0.8\()^19\) + (20 choose 2) * 0.\(2^2\) * (0.8\()^18\)

Using a calculator, we get:

P(X < 3) ≈ 0.1792

c. To find the probability that none of the contracts experience cost overruns, we use the binomial probability formula:

P(X = 0) = (20 choose 0) * 0.2^0 * (0.8)^20

Using a calculator, we get:

P(X = 0) ≈ 0.0115

d. The mean number of contracts that experience cost overruns is given by the formula:

μ = n*p

where "n" is the number of contracts sampled (20) and "p" is the probability of a cost overrun (0.2). Thus, we have:

μ = 20 * 0.2

μ = 4

e. The standard deviation of the number of contracts that experience cost overruns is given by the formula:

σ = sqrt(np(1-p))

Plugging in the values, we get:

σ = sqrt(200.2(1-0.2))

σ ≈ 1.79

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