Answers to these pls someone

Answer:

[tex]x^2+\left(b^3-3bc\right)x+c^3=0[/tex]

[tex]\alpha = 1 \quad \textsf{and} \quad \beta=2[/tex]

[tex]b=-3[/tex]

Step-by-step explanation:

If α and β are roots of x² + bx + c = 0 then the equation must be:

[tex](x -\alpha)(x - \beta) = 0[/tex]

Expanding we get:

[tex]x^2-\beta x - \alpha x + \alpha \beta = 0[/tex]

[tex]x^2-(\alpha +\beta ) x + \alpha \beta = 0[/tex]

Equating the coefficients we get:

[tex]\alpha + \beta=-b[/tex]

[tex]\alpha \cdot \beta=c[/tex]

For an equation with the roots α³ and β³, the sum of the roots can be rewritten in terms of (α + β) and (α·β) using the Sum of Cubes formula, and the Square of Binomials formula:

[tex]\begin{aligned}\alpha^3+\beta^3&=(\alpha + \beta)(\alpha^2-\alpha\beta+\beta^2)\\&=(\alpha + \beta)(\alpha^2+2\alpha\beta+\beta^2-3\alpha\beta)\\&=(\alpha + \beta)((\alpha+\beta)^2-3\alpha\beta))\end{aligned}[/tex]

Substitute in the expressions for (α + β) and αβ to find the sum of the roots α³ and β³ in terms of b and c:

[tex]\begin{aligned}\alpha^3+\beta^3&=(\alpha + \beta)((\alpha+\beta)^2-3\alpha\beta))\\&=(-b)((-b)^2-3c))\\&=-b(b^2-3c)\\&=-b^3+3bc\end{aligned}[/tex]

The product of the roots α³ and β³ in terms of c is:

[tex]\alpha \cdot \beta = c[/tex]

[tex](\alpha \cdot \beta)^3 = c^3[/tex]

[tex]\alpha^3 \cdot \beta^3=c^3[/tex]

For a quadratic equation in the form x² + bx + c = 0:

So for x² + bx + c = 0 with roots α³ and β³:

[tex]x^2-(\alpha^3 + \beta^3)x+(\alpha^3 \cdot \beta^3)=0[/tex]

[tex]x^2-\left(-b^3-3bc\right)x+c^3=0[/tex]

[tex]x^2+\left(b^3-3bc\right)x+c^3=0[/tex]

Therefore, the equation with the roots α³ and β³ is:

[tex]\boxed{x^2+\left(b^3-3bc\right)x+c^3=0}[/tex]

Substitute the given value of c = 2:

[tex]x^2+\left(b^3-3b(2)\right)x+2^3=0[/tex]

[tex]x^2+\left(b^3-6b\right)x+8=0[/tex]

If b³ - 6b + 9 = 0, then (b³ - 6b) = -9.

Substitute this into the equation:

[tex]x^2-9x+8=0[/tex]

Factor:

[tex](x-1)(x-8)=0[/tex]

Therefore, the roots of the equation with the roots α³ and β³ are 1 and 8, so the values of α and β are:

[tex]\alpha^3 =1 \implies \alpha = 1[/tex]

[tex]\beta^3=8 \implies \beta=2[/tex]

To find the real roots of b³ - 6b + 9 = 0, substitute the found values of α and β into the expression for b:

[tex]b=-(\alpha + \beta)[/tex]

[tex]b=-(1+2)[/tex]

[tex]b=-3[/tex]

Therefore, the real root of b³ - 6b + 9 = 0 is b = -3.

We can confirm this by substituting b = -3 into the equation:

[tex]\begin{aligned}(-3)^3-6(-3)+9&=-27+18+9\\&=-9+9\\&=0\end{aligned}[/tex]

An equation, in slope-intercept form, for the line of best fit for the table  above is y = 0.05x + 0.7.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (1.3 - 1.1)/(4 - 8)

Slope (m) = -0.2/4

Slope (m) = 0.05

At data point (8, 1.1) and a slope of 0.05, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 1.1 = 0.05(x - 8)  

y = 0.05x - 0.4 + 1.1

y = 0.05x + 0.7

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Becky is correct if x + y = 180° ,   indicating that BCDF is a cyclic quadrilateral. Teboho's claim would be valid if x +  y ≠ 180°.

A cyclic quadrilateral, also known as an inscribed quadrilateral in Euclidean  geometry, is a quadrilateral with all   of its vertices on a single circle.

To determine the size of angle C in terms of x and y, we can use the fact that tangent BC touches the circle ABDE at B.

This means that angle C is equal to the angle formed between the chord AD and the tangent BC at point B.

4.1.1

Since AB || EC, we can use the alternate interior angles theorem to determine the relationship between angles C and A₁.

Angle C is equal to angle A₁ because they are corresponding angles.

Therefore, the size of angle C is y.

4.1.2   - To determine if BCDF is a cyclic quadrilateral

 In order to check if BCDF is cyclic, we need to determine if the opposite angles in the quadrilateral add up to 180° .

Let's consider angle BCD and angle BFD:

Angle BCD is equal to angle B₁ (given) and is represented by x.

Angle BFD is equal to angle A₁ (since BCDF is a cyclic quadrilateral) and is represented by y.

If x + y = 180 degrees, then BCDF is a cyclic quadrilateral.

conclusively , Becky is correct if x + y = 180°, indicating that BCDF is a cyclic quadrilateral. Teboho's claim would be valid if x + y ≠ 180°.

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Answer:

Shown below.

Step-by-step explanation:

The commutative property is a property of binary operations that states that the order of the operands does not affect the result of the operation. In other words, if you swap the order of the operands, you will still get the same result. This property holds true for addition and multiplication but not for subtraction and division.

Therefore, the equivalent expressions in order are 7 × 5, 24 + 10, 1 × 3 and 5 × 7.

Answer:

-------------------------

Consider the properties of absolute value:

Since x is negative, 2x will also be negative.

To remove the absolute value symbol and keep the expression equivalent, multiply by -1:

The value of absolute value symbol is,

⇒ - 2x

Since, Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We have to given that;

⇒ |2x|

If x < 0

Hence, Number write in without the absolute value symbol as;

⇒ |2x|

⇒ - 2x

Thus, The value of absolute value symbol is,

⇒ - 2x

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Answer:

The sample size is 30.

The sample mean is $36,900.

The sample standard deviation is $8,250.

The confidence level is 98%.

The critical value for a 98% confidence interval with 29 degrees of freedom is 2.457.

The confidence interval is calculated as follows:

(Sample mean - t-critical value * sample standard deviation / sqrt(sample size), Sample mean + t-critical value * sample standard deviation / sqrt(sample size))

(36,900 - 2.457 * 8,250 / sqrt(30), 36,900 + 2.457 * 8,250 / sqrt(30))

(21,668.25, 52,131.75)

Therefore, a 98% confidence interval for the mean account valuation of the population of customers is $21,668.25 to $52,131.75.

Step-by-step explanation:

The number 24, 22, 34, 28, 29, 24, 25, 29, a and b have a median of 27 and a mode of 29 then a is 26 and b is 29

We need to arrange the numbers in order to find the median

22, 24, 24, 25, 28, 29, 29, 34, a, b

The middle number when the numbers are arranged in order is the median

Since we have 10 numbers, the median is the average of the 5th and 6th numbers:

Median = (28 + 29)/2 = 27

The mode is the most occuring number.

Here, the number 29 appears twice, which is more than any other number.

Since a < b, we know that a must be one of the numbers that appears earlier in the list.

Since the mode is 29, we know that b must be 29.

Looking at the list, the only number earlier than 28 that is not 24, 25, or 22 is 26.

Hence, the value of a is 26 and b is 29

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The answer is B. Data set A in not linear, and data set B is not linear.

Answer:

A and C

Step-by-step explanation:

A. 52 mph and C. 48 mph represent the same level of accuracy compared to the speed limit sign since they are within the range of -5 mph to +5 mph from the limit of 55 mph. Therefore, the correct answers are A and C.

Yes, there are two possible shapes that fit this description.

Lilly is describing a parallelogram, which is a quadrilateral with two pairs of parallel opposite sides.

A parallelogram has opposite sides that are congruent and parallel, and opposite angles that are congruent. Therefore, it has two pairs of equal opposite sides.

There are two types of parallelograms: a rectangle and a rhombus. A rectangle is a parallelogram with four right angles, while a rhombus is a parallelogram with four congruent sides.

Both shapes have two pairs of equal opposite sides and opposite sides that are parallel.

Therefore, Robin is correct that there are two possible shapes. Depending on whether all four angles are right angles or all four sides are congruent, the shape could be a rectangle or a rhombus. It is also possible for a shape to be both a rectangle and a rhombus, in which case it would be called a square.

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Step-by-step explanation:

To find f(8 months), we can use the formula:

f(n) = 6000 - 120n

Substituting n = 8, we get:

f(8) = 6000 - 120(8) = 5160

Therefore, f(8 months) is 5160.

To find f(22 months), we can again use the formula:

f(22) = 6000 - 120(22) = 3480

Therefore, f(22 months) is 3480.

To find f(4 years), we need to convert 4 years to months and then use the formula:

f(4 years) = f(48) = 6000 - 120(48) = 48

Therefore, f(4 years) is 48.

Answer to Khan's questions:

- Is the input in savings dollars? Yes, the initial account balance is given in dollars.

- Will the account be at $4660 in one year? No, we cannot determine this without additional information about the interest rate or other factors that may affect the account balance.

- Will it take 50 months for the account to hit $0? No, we cannot determine this without additional information about the interest rate or other factors that may affect the account balance.

When we have two large independent samples from two populations, the difference in the two sample means will follow an approximate normal distribution.

This is due to the central limit theorem, which states that when we have a large enough sample size (typically, greater than 30), the distribution of the sample means will be normal regardless of the underlying distribution of the populations. As the two samples are independent, the difference in means will also be independent. Therefore, we can use the normal distribution to estimate the probability of observing a certain difference in means. It is important to note that this approximation may not hold for small sample sizes, and the underlying distributions of the populations can also impact the shape of the distribution. We can say that the normal distribution approximation is widely used in statistics and hypothesis testing, making it an essential concept for understanding and interpreting data.

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The interquartile range for the museum is 280

The interquartile range for the farm is 200

The interquartile range for the museum is greater than the interquartile range for the farm.

This suggests that the numbers of visitors to the museum are less consistent.

The interquartile range is solved using the formula

= top quartile - bottom quartile

Where

the bottom quartile is at the box's edge, the top side.

the top quartile is towards the box's edge the downside.

the museum

The interquartile range is

= top quartile - bottom quartile

= 500  -  220

= 280

the farm

The interquartile range is

= top quartile - bottom quartile

= 460  -  260

= 200

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Nina will catch up Kevin in 1 hour.

Given that Kevin has completed 90 miles in the race and Nina completed 15 miles less, Kevin continues the race at a constant rate of 25mph and Nina continues the race at a constant rate of 30mph.

So, Nina covered the distance = 90-15 = 75 miles.

So,

When will finish the race, Kevin has already covered 15 miles,

So, Nina needs to catch Kevin by covering 15 + 15 = 30 miles

So, time taken by Nina cover 30 miles = 30/30 = 1 hours

Hence Nina will catch up Kevin in 1 hour.

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The length of the segment NO is 17.9

The given parameters are;

Length of side, QP = 9

Length of QO = 14

The length of  segment NO we have to find.

NO² = qo×(qp×qo)

Plug in the values of qo, qp

Which gives NO² =14(9×14)

=322

NO=17.9

Hence, the length of the segment NO is 17.9

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Given the circle below with tangent \overline{NO} NO and secant \overline{QPO} QPO ​ . If QP=9 and QO=14, find the length of \overline{NO} NO . Round to the nearest tenth if necessary. Q

The value of n is -77/24.

To solve this expression,

Start with the expression: 3/5 × (-6 + 2 2/3) ÷ 1 1/2 - 1 1/8 - 5/2 + (1 4/5 ÷ 9/10).

Simplify the addition and subtraction within parentheses first:

-6 + 2 2/3 = -6 + 8/3 = -18/3 + 8/3 = -10/3

1 4/5 ÷ 9/10 = 9/5 ÷ 9/10 = 9/5 × 10/9 = 90/45 = 2

Now the expression becomes: 3/5 × (-10/3) ÷ 1 1/2 - 1 1/8 - 5/2 + 2.

Next, let's simplify the division:

-10/3 ÷ 1 1/2 = -10/3 ÷ 3/2 = -10/3 × 2/3 = -20/9

The expression now becomes: 3/5 × (-20/9) - 1 1/8 - 5/2 + 2.

Now let's simplify the subtraction:

1 1/8 = 8/8 + 1/8 = 9/8

The expression becomes: 3/5 × (-20/9) - 9/8 - 5/2 + 2.

Convert all fractions to a common denominator:

The common denominator for 5, 9, 8, and 2 is 360.

The expression becomes: (3/5) × (-160/72) - (9/8) × (45/45) - (5/2) × (180/180) + (2/1) × (180/180).

Perform the multiplications:

(3/5) × (-160/72) = -480/360 = -4/3

(9/8) × (45/45) = 405/360 = 9/8

(5/2) × (180/180) = 900/360 = 5/2

(2/1) × (180/180) = 360/360 = 1

The expression becomes: (-4/3) - (9/8) - (5/2) + 1.

Simplify the expression:

-4/3 - 9/8 - 5/2 + 1 = (-128/96) - (27/24) - (60/48) + (48/48) = -128/96 - 108/96 - 120/96 + 48/48

= (-128 - 108 - 120 + 48)/96 = -308/96 = -77/24.

Therefore, the value of n is -77/24.

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True

False

True

1. True, the legs of a right triangle are connected to the right angle.

2. False, the hypotenuse of a right triangle is a specific side that is opposite to the right angle, and it is always the longest side of the triangle. The other two sides are called legs.

3. True, the side opposite to the right angle in a right triangle is called the hypotenuse.

The volume of solids, to the nearest tenth are calculated as: 7. 103.7 km³; 18. 5575.3 cubic cm; 19. 2144.7 mi³

The first shape is a cone while the other two solids are spheres, therefore:

The volume of a sphere (V) = 4/3 * πr³

Volume of cone (V) = 1/3 * πr²h

17. radius (r) = 6/2 = 3 km

height (h) = 11 km

Volume = 1/3 * π * 3² * 11 ≈ 103.7 km³

18. The solid is a sphere with a diameter of 22 cm. Therefore, we have:

radius (r) = 22/2 = 11 cm

Volume of the sphere = 4/3 * π(11)³

Volume of the sphere ≈ 5575.3 cubic cm

19. r = 16/2 = 8 mi

Volume = 4/3 * π * 8³ ≈ 2144.7 mi³

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Question:

Find the Volume of each solid. Round to the nearest tenth is necessary.

The function that provides the height of the stack of cups based on any number of cups is height(n) = 7.5 + 1.5(n-1)

Let's assume that the height of one cup is 7.5 inches.

When we add the second cup, the bottom of the second cup rests on the top of the first cup, including the 1.5" lip.

Therefore, the height of the second cup is 7.5 inches plus 1.5 inches, or 9 inches.

When we add the third cup, the bottom of the third cup rests on the top of the second cup, including the 1.5" lip.

Therefore, the height of the third cup is the height of the first two cups plus 1.5 inches, or 16.5 inches.

The function we get by continuing this pattern for any number of cups n by using the formula:

height(n) = 7.5 + 1.5(n-1)

Therefore, the function that provides the height of the stack of cups based on any number of cups is height(n) = 7.5 + 1.5(n-1)

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Therefore, in a 2x2 table, the pattern of cell frequencies that would indicate independence is d. All cell frequencies are exactly the same.

In a 2x2 contingency table, the expected cell frequencies under the null hypothesis of independence are equal for all cells. If the observed cell frequencies in the table are approximately equal to the expected cell frequencies, then we can conclude that there is no significant association between the two variables being studied. In other words, the pattern of observed cell frequencies is consistent with the null hypothesis of independence. Therefore, if all cell frequencies are exactly the same, it suggests that the variables are independent, as each cell has an equal chance of being filled by any observation regardless of the value of the other variable.

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The simplified expression in this problem has the result given as follows:

40.

The expression for this problem is defined as follows:

[tex]\frac{(8^4)^3 \times 5^2 \times 5^3}{8^7 \times (8 \times 5)^4}[/tex]

Applying the power of power rule, which states that we keep the bases and multiply the exponents we, have that:

Hence:

[tex]\frac{8^{12} \times 5^2 \times 5^3}{8^7 \times 8^4 \times 5^4}[/tex]

When two terms with the same base and different exponents are multiplied, we keep the bases and add the exponents, hence:

[tex]\frac{8^{12} \times 5^5}{8^{11} \times 5^4}[/tex]

When two terms with the same base and different exponents are divided, we keep the bases and add the exponents, hence:

8 x 5 = 40.

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We know that the original recipe is for 6 people, and it requires 800g of pineapple, 4 egg whites, and 1/2 lemon.

If Dan uses 2 1/2 lemons instead of 1/2 lemon, that means he's using 5 times as much lemon juice. Therefore, he's likely making 5 times the original recipe.

So, for Dan's pineapple sorbet, he would need:

800g x 5 = 4000g (or 4kg) of pineapple

4 x 5 = 20 egg whites

1/2 lemon x 2.5 = 1 1/4 lemons (but since we can't really use a quarter of a lemon, we'll just say he needs 1.5 lemons)

Therefore, Dan is making pineapple sorbet for 6 x 5 = 30 people.

The best measure of central tendency for Patrick's scores would be the median, which is 72.5.

To determine which measure of central tendency best describes Patrick's scores, let's consider the three main measures: mean, median, and mode.
1. Mean: Calculate the average by adding all scores and dividing by the number of tests.
(70 + 74 + 72 + 71 + 73 + 96) / 6 = 456 / 6 = 76
2. Median: Arrange the scores in ascending order and find the middle value.
70, 71, 72, 73, 74, 96
There are two middle values (72 and 73), so the median is their average: (72 + 73) / 2 = 72.5
3. Mode: Identify the most frequent score.
There is no mode, as all scores occur only once, except for the outlier (96).
The best measure of central tendency for Patrick's scores would be the median, which is 72.5. The median is less influenced by the outlier (96) and provides a more accurate representation of Patrick's typical performance on the science tests.

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Answer:

the answer would be b and d

Step-by-step explanation:

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Using the combination formula, we calculate that there are 133,784,560 different 7-card hands that can be chosen from a standard 52-card deck.

A standard 52-card deck has 52 unique cards, and to find the number of different 7-card hands that can be chosen, we use the combination formula, which is C(52,7). This formula represents the number of ways to choose 7 cards out of a total of 52 cards, where order does not matter.

The formula for C(52,7) is derived by using factorials. For example, 52! represents 52 x 51 x 50 x 49 x 48 x 47 x 46 x ... x 2 x 1, which is the total number of ways to arrange all 52 cards. The denominator, 7! x 45!, represents the number of ways to arrange the 7 chosen cards out of 52, and also the number of ways to arrange the remaining 45 cards that were not chosen.

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The formula for the volume of a rectangular prism is [tex]V = whl[/tex], where [tex]V[/tex] is the volume, [tex]w[/tex] is the width, [tex]h[/tex] is the height, and [tex]l[/tex] is the length. We can calculate the volume by substituting our values into this formula and solving.

[tex]V = 12 \cdot 2 \cdot 3\\V = 72[/tex]

So, our answer is [tex]\boxed{72 \text{ ft}^2}[/tex].

If the question is asking you for the number of cubes that we can fit, we replace each length with cubes that are [tex]1 \cdot 1 \cdot 1 \text{ ft}^3[/tex], which are the largest cubes you can fit without overlap, you fit 72 cubes in the prism, each of length [tex]1 \text{ ft}[/tex].