3. Consider the quadratic equation x2 + 2x - 35 = 0. Solve by factoring and using the zero-product property. What are solutions to quadratic equations called? Show your work.

Using the given information, we found the angles of the right triangle as:

∠R = 56.30º

∠P = 33.7º

∠Q = 90º

What is a right triangle?

A triangle with a right angle, also known as a right-angled triangle or right-angled triangle, more technically an orthogonal triangle, has two perpendicular sides. The foundation of trigonometry is the relationship between the sides and various angles of the right triangle. A right triangle's hypotenuse is its longest side, its "opposite" side is the one that faces a certain angle, and its "adjacent" side is the one that faces the angle in question. To describe the sides of right triangles, we utilise specific terminology. The side opposite the right angle is always the hypotenuse of a right triangle.

Given one of the angles of the triangles as 90º.

So it is a right triangle.

PQ = 7.5cm
QR = 5cm
∠Q = 90º

The figure is given below.

Now PR is the hypotenuse.

Hypotenuse = [tex]\sqrt{base^2+height^2}[/tex]

base = 5 = QR

height = 7.5 = PQ

PR = [tex]\sqrt{5^2+7.5^2} = \sqrt{81.25}[/tex] = 9.01 cm

We can find the angles using trigonometric relations.

sin R = PQ/PR = 7.5/9.01 = 0.832

∠R = 56.30º

∠P = 180 - (∠Q+∠R) = 180 - (90+56.30) = 33.7º

Therefore using the given information, we found the angles of the right triangle as:

∠R = 56.30º

∠P = 33.7º

∠Q = 90º

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Step-by-step explanation:

use the function sohcahtoa to solve as all these triangles are right angled

0.3406 (approx)

Given that 60% of all Americans live in cities with population greater than 100,000 people. If 47 Americans are randomly selected, we need to find the probability thata. Exactly 25 of them live in cities with population greater than 100,000 people.b. At most 26 of them live in cities with population greater than 100,000 people.c. At least 30 of them live in cities with population greater than 100,000 people.d. Between 23 and 28 (including 23 and 28) of them live in cities with population greater than 100,000 people.Probability is defined as the ratio of the favorable outcomes to the total outcomes of an event. The formula to calculate probability is given by;Probability = Number of favorable outcomes / Total number of outcomesa. Exactly 25 of them live in cities with a population greater than 100,000 people.Probability of exactly 25 Americans living in cities with population > 100,000 people is given by the probability mass function of the binomial distribution.P( X = 25) = 47 C 25 * (0.6)25 * (0.4)22= 0.1213 (approx)b. At most 26 of them live in cities with population greater than 100,000 people.We need to find the probability of at most 26 Americans living in cities with population greater than 100,000 people. This means the number of Americans living in cities with population greater than 100,000 is 0, 1, 2, ..., 26.P(X ≤ 26) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 26)P(X ≤ 26) = ΣP(X = r)Where r varies from 0 to 26P(X ≤ 26) = Σ (47 C r * (0.6)r * (0.4)47-r )r = 0 to 26= 0.6413 (approx)c. At least 30 of them live in cities with population greater than 100,000 people.We need to find the probability of at least 30 Americans living in cities with population greater than 100,000 people. This means the number of Americans living in cities with population greater than 100,000 is 30, 31, 32, ..., 47.P(X ≥ 30) = P(X = 30) + P(X = 31) + P(X = 32) + ... + P(X = 47)P(X ≥ 30) = ΣP(X = r)Where r varies from 30 to 47P(X ≥ 30) = Σ (47 C r * (0.6)r * (0.4)47-r )r = 30 to 47= 0.0031 (approx)d. Between 23 and 28 (including 23 and 28) of them live in cities with population greater than 100,000 people.We need to find the probability of the number of Americans living in cities with population greater than 100,000 is between 23 and 28 (both inclusive).P(23 ≤ X ≤ 28) = P(X = 23) + P(X = 24) + ... + P(X = 28)P(23 ≤ X ≤ 28) = ΣP(X = r)Where r varies from 23 to 28= Σ (47 C r * (0.6)r * (0.4)47-r )r = 23 to 28= 0.3406 (approx)Hence, the required probabilities are,a. P(X = 25) = 0.1213 (approx)b. P(X ≤ 26) = 0.6413 (approx)c. P(X ≥ 30) = 0.0031 (approx)d. P(23 ≤ X ≤ 28) = 0.3406 (approx)

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Answer:

Step-by-step explanation: m miles

The new mural dimensions are 267 inches by 273 inches.To find the new dimensions, we must add 33 inches to the original length of 234 inches, giving us a new length of 267 inches.

To calculate the new dimensions of Rhiannon's mural, we must first identify the original dimensions of the mural which are 234 inches by 245 inches. To find the new dimensions, we must add 33 inches to the original length of 234 inches, giving us a new length of 267 inches. We must also add 28 inches to the original width of 245 inches, giving us a new width of 273 inches. Therefore, the new dimensions of Rhiannon's mural are 267 inches by 273 inches.

The complete question is :

A rectangle mural measures 234 inches by 245. Rhiannon creates a new mural that is 33 inches longer. What are the dimensions of Rhiannon's new mural?

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The value of a is -9

When expressing a polynomial in its standard form, the greatest degree of terms are written first, followed by the next degree, and so on.

[tex]P(x)=4x^5+3x^2+2x+a[/tex]

Question might be asking to find the value of 'a'  at the point (1, 0) on the graph of  [tex]P(x)=4x^5+3x^2+2x+a[/tex]

Substitute the point into the polynomial.  i.e., x=1, y=0

=>  [tex]0=4(1)^5+3(1)^2+2(1)+a[/tex]

=> 0=  4*1 + 3*1+2 +a

=> 0= 4+3+2+a

=> 0=9 +a

=> a= -9

The value of a is -9

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Complete Question:

Find the value of  [tex]P(x)=4x^(5)+3x^(2)+2x+a[/tex]

Answer: x=10/9

Step-by-step explanation:

Answer: Let's use algebra to check if Tim is correct.

Let x be the original price of the jeans.

When the jeans are marked down by 20%, the new price becomes:

x - 0.2x = 0.8x

Tim claims that if we increase the new price by 20%, we will get the original price:

0.8x + 0.2(0.8x) = x

Simplifying the left-hand side:

0.8x + 0.16x = x

0.96x = x

This is not true for all values of x, which means that Tim's claim is not correct.

Let's use the given information that the original price is $200 to find out what the new price is after the 20% discount:

New price = Original price - Discount

New price = $200 - 20% of $200

New price = $200 - $40

New price = $160

If we increase the new price of $160 by 20%, we get:

New price + 20% of new price

$160 + 20% of $160

$160 + $32

$192

This is not equal to the original price of $200, so we can conclude that Tim's claim is incorrect.

Answer:

It's very difficult

maby

If a store sells two types of shirts. The number of short-sleeved shirts that were sold than long-sleeved shirts is 24 shirts.

Number of short-sleeved shirts sold =S

Number of long-sleeved shirts sold = L

we know that:

s + l = 48 (equation 1) -- Total number of shirts sold

12s + 16l = 624 (equation 2) -- Total cost of shirts sold

To solve for the number of short-sleeved shirts sold, we can use equation 1 to express "l" in terms of "s":

l = 48 - s

Substituting this into equation 2

12s + 16(48 - s) = 624

Simplifying and solving for "s"

12s + 768 - 16s = 624

-4s = -144

s = 36

Therefore, 36 short-sleeved shirts were sold.

To find the number of long-sleeved shirts sold, we can use equation 1 again:

s + l = 48

36 + l = 48

l = 12

Therefore, 12 long-sleeved shirts were sold.

How many MORE short-sleeved shirts were sold than long-sleeved shirts,:

36 - 12 = 24

So, 24 MORE short-sleeved shirts were sold than long-sleeved shirts.

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Answer:

QU is 21, and I *think* that VS would be 8.5.

Answer:

QV = 21 , VS = 8.5

Step-by-step explanation:

QU and RS are medians of Δ PQR

the point V where the medians intersect is the centroid.

on each median the distance from the vertex to the centroid is twice as long as the distance from the centroid to the midpoint.

then

VU = [tex]\frac{1}{2}[/tex] QV = [tex]\frac{1}{2}[/tex] × 14 = 7

so

QU = QV + VU = 14 + 7 = 21

and

VS = [tex]\frac{1}{2}[/tex] RV = [tex]\frac{1}{2}[/tex] × 17 = 8.5

The ratio x:y in its simplest form is 7:4.

The ratio x:y can be found by dividing x by y. In this case, we can start with the equation y = 4x/7 and solve for x in terms of y.

Multiplying both sides by 7, we get:

7y = 4x

Dividing both sides by 4, we get:

x = 7y/4

Now we can substitute this expression for x into the ratio x:y:

x:y = (7y/4):y

Simplifying by canceling out the y, we get:

x:y = 7/4

So the ratio x:y in its simplest form is 7:4.

This means that for every 7 units of x, there are 4 units of y. The ratio cannot be simplified further because 7 and 4 do not have any common factors other than 1.

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Answer: -7.6n + 11.1

Step-by-step explanation:

We will simplify the given expression by combining like-terms.

   Given:

3.1 - 4.6n - 3n + 8

  Subtract like-terms:

3.1 - 7.6n + 8

   Add like-terms:

11.1 - 7.6n

Answer:

[tex] \sf \: -7.6n + 11.1 [/tex]

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The expression is,

→ 3.1 - 4.6n - 3n + 8

Let's simplify the expression,

→ 3.1 - 4.6n - 3n + 8

→ -4.6n - 3n + 3.1 + 8

→ (-4.6n - 3n) + (3.1 + 8)

→ (-7.6n) + (11.1)

→ -7.6n + 11.1

Hence, the answer is -7.6n + 11.1.

Based on the coordinate plane, the ordered pair for the post office is (9, 9).

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.

By critically observing the coordinate plane shown in the image attached below, we can logically deduce that the coordinates of the post office would be located in quadrant I at point (9, 9).

Similarly, the the coordinates of the library is also located in quadrant I and it is represented by the ordered pair (6, 2).

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The correct statement regarding the translation of the functions f(x) and h(x) is given as follows:

The graph of h is a translation of 4 units left and 7 units down of f(x).

A translation happens when either a figure or a function are moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The meaning of each translation from function f(x) to function h(x) is given as follows:

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The measure of angle ABD as required to be determined from the task content is; 32°.

As evident in the task content;

m<ABD + m<CBD = m<ABC = 90°.

This follows from the fact that the angle ABC is a right angle.

4x - 4 + 2x + 40 = 90

6x = 54

x = 54 / 6 = 9

Ultimately, the measure of angle ABD is; 4(9) - 4 = 36 - 4 = 32.

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The given equation x² + 12x − 1  can be written in the form of algebraic identity (x + 6)² -37

Algebraic identities are equations in algebra that hold regardless of the value of each of their variables. The factorization of polynomials makes use of algebraic identities. On both sides of the equation, they have variables and constants.

Write x² + 12x − 1 in the form (x + a)² + b, where a and b are constants.

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added to a variable before being multiplied by it.

[tex]The \ given \ equation\ is\ \\\\x^2+12x-1\\let , \ x^2+12x=1\\\\add \ both\ side \ \frac{12^2}{4}\\\\x^2+12x+\frac{144}{4}=1+\frac{144}{4}\\\\x^2+12x+36=1+36\\x^2+2.6.x+6^2-37\\\\(x-6)^2-37\\\\[/tex]

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Answer:

25&36

Step-by-step explanation:

25 is a square number and so is 36 and there product is 900. They are also consecutive.

Just assemble the square numbers from the least i.e 4 and try solving as asked to see if it gives 900.

Thus you'll land on 25&36

Answer:

5² and 6²

Step-by-step explanation:

Use trial and error

Start from the product of 1² and 2²

1×2²

1×4=4

4<900 (incorrect)

2²×3²

4×9=36

36<900(incorrect)

3²×4²

9×16=144

144<900(incorrect)

4²×5²

16×25=400

400<900(incorrect) *but close

5²×6²

25×36 =900

900=900(correct)

: . 5² and 6² are the two consecutive square numbers.

Answer:

Population: The population is the total demand (sale) of the product over all days.

Variable of interest: The variable of interest is the daily demand (sale) of the product.

To obtain the confidence interval for the average daily demand at a confidence level of 97%, we can use the following formula:

Confidence interval = sample mean ± (t-value x standard error)

where t-value is the value from the t-distribution for the desired confidence level and degrees of freedom, and the standard error is calculated as:

standard error = sample standard deviation / √n

where n is the sample size.

Using the given data, we can calculate:

Sample mean = (35+44+38+55+33+56+60+45+48+40+45+35+42)/12 = 44.5

Sample standard deviation = 9.92

Degrees of freedom = n-1 = 12-1 = 11

From the t-distribution table with 11 degrees of freedom and a confidence level of 97%, the t-value is approximately 2.718.

Therefore, the confidence interval for the average daily demand is:

Confidence interval = 44.5 ± (2.718 x 9.92/√12) = 44.5 ± 9.14

The lower limit is 44.5 - 9.14 = 35.36 and the upper limit is 44.5 + 9.14 = 53.64.

So, we can say with 97% confidence that the true population average daily demand falls within the range of 35.36 to 53.64 thousand units.

There are 6 + 9 + 9 + 12 + 9 + 9 + 6 = 60 outcomes where the sum of the three dice is greater than 5.

So, the probability of rolling a sum greater than 5 is given by:

Probability = Number of favorable outcomes / Total number of outcomes
Probability = 60 / 216
Probability = 5 / 18

We know that when a dice is rolled, the numbers that come up on the dice are 1, 2, 3, 4, 5 and 6. Since there are three dice, the total number of possible outcomes when they are tossed is given by 6 * 6 * 6 = 216.

Now we have to find the probability of rolling a sum greater than 5. To find this probability, we need to consider all the cases where the sum of the three dice is greater than 5.

The possible outcomes where the sum of the three dice is greater than 5 are:

Sum of 6: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) (Total 6)
Sum of 7: (1, 2, 4), (1, 4, 2), (2, 1, 4), (2, 4, 1), (4, 1, 2), (4, 2, 1), (1, 3, 3), (3, 1, 3), (3, 3, 1) (Total 9)
Sum of 8: (1, 2, 5), (1, 5, 2), (2, 1, 5), (2, 5, 1), (5, 1, 2), (5, 2, 1), (3, 3, 2), (3, 2, 3), (2, 3, 3) (Total 9)
Sum of 9: (1, 3, 5), (1, 5, 3), (3, 1, 5), (3, 5, 1), (5, 1, 3), (5, 3, 1), (4, 2, 3), (4, 3, 2), (2, 4, 3), (3, 4, 2), (2, 3, 4), (3, 2, 4) (Total 12)
Sum of 10: (1, 4, 5), (1, 5, 4), (4, 1, 5), (4, 5, 1), (5, 1, 4), (5, 4, 1), (2, 4, 4), (4, 2, 4), (4, 4, 2) (Total 9)
Sum of 11: (1, 5, 5), (5, 1, 5), (5, 5, 1), (2, 5, 4), (2, 4, 5), (4, 5, 2), (4, 2, 5), (5, 4, 2), (5, 2, 4) (Total 9)
Sum of 12: (3, 4, 5), (3, 5, 4), (4, 3, 5), (4, 5, 3), (5, 3, 4), (5, 4, 3) (Total 6)

Therefore, there are 6 + 9 + 9 + 12 + 9 + 9 + 6 = 60 outcomes where the sum of the three dice is greater than 5.

So, the probability of rolling a sum greater than 5 is given by:

Probability = Number of favorable outcomes / Total number of outcomes
Probability = 60 / 216
Probability = 5 / 18

Hence, the probability of rolling a sum greater than 5 is 5 / 18.

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The simplified expression is 8x² + 24x.

The distributive property of multiplication over addition is a fundamental property of arithmetic that relates multiplication and addition. It states that when you multiply a number by the sum of two or more numbers, you can first distribute the multiplication over each addend and then perform the addition.

In other words, if a, b, and c are any numbers, then:

a x (b + c) = (a x b) + (a x c)

To simplify the expression 4x° (2x + 6)°, we can use the distributive property of multiplication over addition.

4x° (2x + 6)°

= 4x° * 2x° + 4x° * 6° (using distributive property)

= 8x² + 24x (simplifying by multiplying)

Therefore, the simplified expression is 8x² + 24x.

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Answer:

13

Step-by-step explanation:

To find the distance between two points in a coordinate plane, we can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Using this formula, we can find the distance between the points (-3, 6) and (-8, -6) as follows:

d = sqrt((-8 - (-3))^2 + (-6 - 6)^2)

= sqrt((-5)^2 + (-12)^2)

= sqrt(25 + 144)

= sqrt(169)

= 13

Therefore, the distance between the two points in simplest radical form is 13.

for (d), the x value has changed, as the (-1) in the series acts as a multiplier and flips the convergence of the original series. This means that when x = -4, the original series converges, but when x = -7, the series in (d) converges. Therefore, the series in (d) converges.

(a) Diverges

(b) Diverges

(c) Converges

(d) Converges: The original series converges when x = -4 and diverges when x = 6. For (a), (b), and (c), the x value remains the same as in the original series, so the convergence or divergence of the series is the same as the original series. However, for (d) the x value has changed. The (-1) in the series acts as a multiplier and flips the convergence of the original series, so the series converges.

The convergence or divergence of a series is determined by the value of x in the series. In this particular case, when x = -4 the original series converges, and when x = 6 it diverges. For (a), (b), and (c), the x value is the same as in the original series, so the convergence or divergence of the series is the same as the original series. However, for (d), the x value has changed, as the (-1) in the series acts as a multiplier and flips the convergence of the original series. This means that when x = -4, the original series converges, but when x = -7, the series in (d) converges. Therefore, the series in (d) converges.

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around 13 jars of paint will the farmer need to buy in order to paint the entire exterior of the silo.

To find the number of jars the rancher required to get, you really want to know the surface area of both the cone and cylinder.

The method for finding the SA of the cone would be area =[tex]3.14 x r^2 + 3.14 x r x sqrt(r^2 + h^2)[/tex].

SA= 1976.0617791 [tex]m^2[/tex].

The method for finding the SA of the cylinder under the cone would be 3.14 x d x (d/2 + h).

A = 3097.6103564 [tex]m^2[/tex]

Then, at that point, you need to add 1976.0617791+3097.6103564 which gives you 5,073.8. (5,073 is the total surface area of the storehouse)

Then, the partition that by 400.

5,073.8/400 = 12.7

He would have to purchase around 13 jars of paint to have the option to paint the whole storehouse.

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The most specific category that this shape can be sorted into is "Triangle."

What is the triangle?

A triangle is a three-sided polygon, which has three vertices. The three sides are connected with each other end to end at a point, which forms the angles of the triangle. The sum of all three angles of the triangle is equal to 180 degrees.

The Venn diagram shows the relationships between different types of triangles.

The Scalene circle represents all the triangles that have no equal sides.

The Isosceles circle represents all the triangles that have two equal sides.

The Equilateral circle represents all the triangles that have three equal sides.

The triangle with a tick mark on each side does not provide enough information to determine whether it is a scalene, isosceles, or equilateral triangle.

Therefore, the most specific category that this shape can be sorted into is "Triangle."

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ECDF=1/100

The value of the empirical cumulative distribution function (ECDF) at the smallest observation in a sample of 100 observations is 1/100.What is the empirical cumulative distribution function (ECDF)?The empirical cumulative distribution function (ECDF) is a graph that depicts the distribution of a given data set. It is used to illustrate the proportion or percentage of data points that fall below a particular value in the distribution. The ECDF can be used to construct a distribution function for data that do not have an existing theoretical distribution.The ECDF value at the smallest observation in a sample of 100 observations is 1/100. This is because the ECDF takes on a value of 0 for all values that are smaller than the smallest observation. Since there are 100 observations in the sample, each observation has a weight of 1/100. Thus, the ECDF value at the smallest observation is 1/100.

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4) Let the polynomial function of degree 4 that has roots 3, 2, 2+ i as its roots be p(x).

So, the required polynomial function is:

p(x)=(x−3)(x−2)(x−(2+i))(x−(2−i))

= (x−3)(x−2)(x^2−(2+i)x−(2−i)x+(2−i)(2+i))

= (x−3)(x−2)(x^2−2x−ix+ix+4)

= (x−3)(x−2)(x^2−2x+4)

= x^4−9x^3+28x^2−36x+24

Thus, the polynomial function of degree 4 that has real coefficients and has 3, 2 and 2+ i as some of its roots is x^4−9x^3+28x^2−36x+24.

Given: x^3−2x^2+36x−72=0 and 6i is a zero of this polynomial function.

So, we can write it as:

x^3−2x^2+36x−72= (x−6i)(x−(−6i))(x−6)

= (x−6i)(x+6i)(x−6)

As this polynomial function has real coefficients, so the imaginary roots occur in conjugate pairs.

Thus, the remaining zeros are:

−6i (as 6i is a zero, so −6i is also a zero due to conjugate pair of complex roots) and 6 (as 6i is one factor of the polynomial function, so x−6i will be its conjugate factor, which will be x+6i. And, the remaining factor will be x−6)

Therefore, the remaining zeros of x^3−2x^2+36x−72 polynomial function are −6i and 6.

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