need some help on some questions

I chose to explore the Angle Sum Theorem because it is a fundamental theorem in geometry, and it has important implications for many geometric proofs and applications. Moreover, the theorem can be easily visualized and understood, making it a great starting point for learning geometry.

The formal definition of the Angle Sum Theorem is: "In any triangle, the sum of the interior angles is equal to 180 degrees."

In simpler terms, the Angle Sum Theorem states that if you add up all the angles inside a triangle, you will always get a total of 180 degrees. This means that no matter how you move the sides of the triangle or what shape it takes, the total measure of the angles will always be the same.

Let's consider a triangle with angles of 30 degrees, 60 degrees, and 90 degrees. According to the Angle Sum Theorem, the sum of the interior angles of any triangle must be 180 degrees.

So, in this case, we have:

30 degrees + 60 degrees + 90 degrees = 180 degrees

Therefore, the Angle Sum Theorem holds true for this example.

I taught the Angle Sum Theorem to my younger brother, and he was fascinated by it. He said that he had never thought about triangles in this way and found it cool that the angles always added up to the same number, no matter what the triangle looked like. He then went on to ask me more questions about triangles and geometry, which I was happy to answer.

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Answer:

5188.8 [cm³].

Step-by-step explanation:

1) V=V[1]+V[2], where V[1] - the volume of the cube with length of its side 10 [cm], V[2] - the volume of the sphere with radius 10 [cm].

[tex]2) \ V=10^3+\frac{4}{3} \pi*10^3=5.188790*1000=5188.8 \ [cm^3][/tex]

By the multiplication principle of counting, the total number of positive perfect square factors of the product is the product of the number of choices for each prime factor, which is $6\times7\times8=336$. Therefore, 336 positive perfect square integers are factors of(2¹⁰ )(3¹² )(5¹⁵)

We can find the number of positive perfect square integers that are factors of the given product by first considering the prime factorization of the product, which is (2¹⁰ )(3¹² )(5¹⁵)

To get a perfect square factor, we need each prime factor to have an even exponent. Therefore, we can choose any even exponent for the prime factors of 2, 3, and 5, respectively, as long as the exponents are not greater than 10, 12, and 15, respectively.

For the factor of 2, we can choose any even exponent from 0 to 10, so there are 6 choices. Similarly, for the factor of 3, we can choose any even exponent from 0 to 12, so there are 7 choices. Finally, for the factor of 5, we can choose any even exponent from 0 to 15, so there are 8 choices.

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On solving the question we can say that Therefore, the coordinates of equation the turning point are: (3/2, -35/2)

In mathematics, an equation is a statement that two expressions are equal. The equation consists of her two sides divided by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the statement "2x + 3" equals the value "9". The goal of solving an equation is to find the values ​​of the variables to make the equation true. Simple or complex equations, regular or nonlinear, and equations involving one or more factors are all possible. For example, the expression "x2 + 2x - 3 = 0" squares the variable x. Lines are used in many areas of mathematics, including algebra, calculus, and geometry.  

[tex]a) -2(x² + 6x) - 8-2(x² + 6x + 9 - 9) - 8-2[(x + 3)² - 9] - 8-2(x + 3)² + 10[/tex]

Therefore, the expression -2x² - 12x - 8 can be written in the form [tex]a(x + b)² + c[/tex] as:

a = -2, b = -3, c = 10

b) The turning point of curve [tex]y = -2x² - 12x - 8[/tex] is at the apex of the parabola. The vertex coordinates are (-b/a, f(-b/a)). where f(x) is the function that defines the curve. In this case:

a = -2, b = -3

Therefore, the x-coordinate of the turning point is:

[tex]-x = -b/a = -(-3)/(-2) = 3/2y=-2(3/2)212(3/2)-8y=-2(9/4)-18-8y=-9/2-26y=-35/2[/tex]

Therefore, the coordinates of the turning point are:

(3/2, -35/2)

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The population will be 482,720 in 405 years using exponential model that doubles its size every 81 years.

A form of growth known as exponential growth occurs when a quantity's rate of expansion is proportionate to its present value. In other words, the amount increases with time at a faster pace. Several natural and artificial processes, including population expansion, compound interest, and the spread of contagious illnesses, exhibit exponential growth.

The population increases using the exponential model given as:

[tex]P(t) = P_0 * 2^{(t/x)}[/tex]

Substituting the values P₀ = 15,085 and x = 81.

[tex]P(405) = 15,085 * 2^{(405/81)}\\P(405) = 15,085 * 2^5\\P(405) = 15,085 * 32\\P(405) = 482,720[/tex]

Hence, the population will be 482,720 in 405 years using exponential model, that doubles its size every 81 years.

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Answer:

Step-by-step explanation:

The inequality in the first equation is x > 8.

The inequality for the second graph is x ≤ -4 since it's a dot.

The bearing of point D from C as required to be determined in the task content is; 315°.

By definition, the bearing of a point is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass with the point. A bearing is characteristically used to represent the direction of one point relative to another point.

The bearing of the point D from C in this case where the angle motion is clockwise starting from the north pole by convention;

The bearing of D from C is; 360 - 45 = 315°.

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It ahs been proved by mathematical induction that the sum of the squares of 4 consecutive integers is an even integer.

To prove that the sum of the squares of 4 consecutive integers is an even integer.

Let x, (x + 1), (x + 2), (x + 3) be the four consecutive integers.

The sum of the squares of these integers are:

S = x² + (x + 1)² + (x + 2)² + (x + 3)²

Expanding this gives us:

S = 4x² + 12x + 14

Simplifying this gives:

S = 2(2x² + 6x + 7)

The number 2x² + 6x + 7 is either even number or odd number.

However, since it is multiplied by 2x² + 6x + 7, the sum will always be an even number.

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let T be the linear transformation, then (T(A)a = (7, 11) where T defined by T(x) = Ax and a = (-1, 2), and A = (1 4 -1 6). [T(f(x))]a = (1, 3) where T defined by T(f(x)) = f(1) + f'(1)x, f(x) = 4-6x+3x^2 and a = (1, 3). (T(A))y = (5 5). [T(f(x))]y = (-1 -4 0).

Let T be the linear transformation defined by T(x) = A x, where A = 1 4 -1 6, and let a be the vector a = (-1, 2). To compute (T(A)a, we have:

T(A)a = Aa = 1 4 -1 6 * (-1) 2

= (1*-1 + 42) (-1-1 + 6*2)

= (7, 11)

Therefore, (T(A)a = (7, 11).

Let T be the linear transformation defined by T(f(x)) = f(1) + f'(1)x, where f(x) = 4 - 6x + 3x^2, and let a = (1, 3). To compute [T(f(x))]a, we have:

f(1) = 4 - 6 + 3 = 1

f'(x) = -6 + 6x

f'(1) = 0

So, T(f(x)) = f(1) + f'(1)x = 1, and [T(f(x))]a = 1 * (1, 3) = (1, 3).

Therefore, [T(f(x))]a = (1, 3).

Let T be the linear transformation defined by T(x, y) = (2x + y, x + 3y). We are given A = (1 3 2 4) and want to compute (T(A)]y.

First, we need to find the matrix of T with respect to the standard basis of R^2:

[T] = [T(1,0)] [T(0,1)] = [2 1] [1 3] = (2 1)

(1 3)

Now, we can compute (T(A)]y using Theorem 2.14:

(T(A)]y = [T]_y[A]_y = [T]_y[1 2] = (5 5)

Therefore, (T(A)]y = (5 5).

Let T be the linear transformation defined by T(p) = p' - p'', where p' and p'' are the first and second derivatives of p, respectively. We are given f(x) = 6 - x + 2x² and want to compute [T(f(x))]y.

First, we need to find the matrix of T with respect to the standard basis of P2 (the space of polynomials of degree at most 2):

[T] = [T(1)] [T(x)] [T(x²)] = [0 -1 2]

[0 0 -2]

[0 0 0]

Now, we need to find the coordinate vector of f(x) with respect to the standard basis of P2:

[f(x)] = [6 -1 2]

Using Theorem 2.14, we can compute [T(f(x))]y:

[T(f(x))]y = [T]_y[f(x)]_y = [T]_y[6 -1 2] = (-1 -4 0)

Therefore, [T(f(x))]y = (-1 -4 0).

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_____The given question is incomplete, the complete question is given below:

For each of the following parts, let T be the linear transformation defined in the corresponding part of Exercise 5 of Section 2.2. Use Theorem 2.14 to compute the following vectors: (a) (T(A)]a, where A = (1 4   -1 6), (b) [T(f(x))]a, where f(x) = 4 - 6x + 3x^2. 1 3, (c) (T(A))y, where A =(1 3   2 4)  (d) [T(F(x))]y, where f(x) = 6 - x + 2x².

Answer:

3:30

Step-by-step explanation:

We know

Jeni put a cake in the oven at 2:30. The cake takes 1 hour to bake.

What time should it be taken out of the oven?

We take

2:30 + 1 = 3:30

So, it should be taken out of the oven at 3:30

The probability of drawing three kings in a row from a standard deck of cards when the drawn card is not returned to the deck each time is 0.000055

The multiplication rule of probability can be used to determine the likelihood of drawing three kings consecutively from a standard deck of cards when the drawn card is not put back into the deck each time.

Since there are four kings in a deck of 52 cards, the likelihood of drawing a king from a standard 52-card deck is 4/52 or 1/13.

There are still 51 cards in the deck after the first king is drawn, and three of them are kings. Therefore, there is a 3/51 chance of drawing another king.

There are 50 cards left in the deck after drawing the second king, and two of them are kings. The likelihood of drawing a third king is therefore 2/50 or 1/25.

Therefore, the probability of drawing three kings in a row from a standard deck of cards when the drawn card is not returned to the deck each time is:

(1/13) x (3/51) x (1/25) = 3/54,600 or approximately 0.000055.

Hence, 0.000055 is the probability of drawing three kings in a row from a standard deck of cards when the drawn card is not returned to the deck each time.

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The simplified value of 3㏑3  is 3.296

An exponent or power to which a given base number must be increased in order to arrive at a certain value is provided by a logarithm, which is a mathematical function.

With a single natural logarithm, the given expression is;

⇒ 3㏑3

we can write it as,

⇒ ㏑ 3³

⇒ ㏑ 27

⇒ 3.2958

Therefor, simplified value of  3㏑3  is 3.296.

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The simplified value of the logarithmic term 3㏑3 is 3.296. Simple logarithm is nothing more than log in base 10.

A logarithm is a mathematical function that specifies the exponent or power to which a given base number must be raised in order to reach a particular value.

Exponent and logarithm are inverses of one another. Logarithm with base e is what is also known as a natural logarithm. Simple logarithm is nothing more than log in base 10.

The provided expression is with a single natural logarithm:

⇒ 3㏑3

we can write it as,

⇒ ㏑ 3³

⇒ ㏑ 27

⇒ 3.2958

Therefore, simplified value of 3㏑3 is 3.296.

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Because g(x) is continuous on the interval, we can see that the correct option is the last one (counting from the top)

Here we have the function g(x), and we know that it is continuous on the interval [1, 6], and that:

g(1) = 18

g(6) = 11

If it is continuous, then g(x) covers all the values between 18 and 11 in the given interval, this means that there must exist a value c in the given interval such that when we evaluat g(x) in that value c, we get the outcome 12, and we know this by the intermediate value theorem.

So the correct optionis the last one.

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The types of quadrilateral are: Rectangle HAEG, ABFE. Trapezium GCDK. b. The properties of quadrilateral required to identify parallelograms are corresponding sides and angles are equal.

A closed shape called a quadrilateral is created by connecting four points, any three of which cannot be collinear. A quadrilateral is a polygon with four sides, four angles, and four vertices, to put it simply. The Latin term "quadra" (which means four) and "Latus" (which means sides) are the roots of the English word "quadrilateral." It should be noted that a quadrilateral's four sides could or might not be equal to one another. There are several kinds of quadrilaterals, and each one is distinguished from the others by its own special characteristics.

The types of quadrilateral in the given window are:

Rectangle HAEG, ABFE.

Trapezium GCDK

Triangle COD, LEF.

The properties of quadrilateral required to identify parallelograms are corresponding sides and angles are equal.

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The two transformations applied to quadrilateral JKLM are rigid transformations, quadrilateral PQRS is congruent to quadrilateral JKLM, and the corresponding sides are proportional but not equal.

The dilation is a transformation in which all points are moved by the same scale factor, but the scale factor is different for each point. In this case, the dilation was centered at the origin and had a scale factor of . This means that all points were moved by the same scale factor of .

Quadrilateral PQRS is the image of quadrilateral JKLM after translating it 6 units down and 6 units to the right, followed by a dilation centered at the origin with a scale factor of . This means that quadrilateral PQRS is congruent to quadrilateral JKLM, and the corresponding sides are proportional but not equal.

In this case, the translation of quadrilateral JKLM was 6 units down and 6 units to the right.

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The statement " when account groups are created, they will determine the valid number interval for each of the groups of general ledger accounts " is  false because it incorrectly states that account groups determine the valid number intervals for each group of general ledger accounts

Account groups in themselves do not determine the valid number intervals for each group of general ledger accounts.

Rather, the number intervals for each group of general ledger accounts are defined by the chart of accounts, which is a structured list of all the accounts used by an organization to define its financial reporting.

Account groups, on the other hand, are used to group together accounts with similar characteristics or usage, and to assign authorizations for access and maintenance of the accounts.

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Answer:

Step-by-step explanation:

We can use the formula for the future value of an annuity to determine how much Marcia needs to invest today to meet her retirement goal of $90,000. The formula for the future value of an annuity is:

FV = PMT x [(1 + r/n)^(n*t) - 1] / (r/n)

where:

FV = future value of the annuity

PMT = payment (or deposit) made at the end of each compounding period

r = annual interest rate

n = number of compounding periods per year

t = number of years

In this case, we want to solve for the PMT (the amount Marcia needs to invest today). We know that:

Marcia wants to retire in 15 years (when she is 65), so t = 15

The interest rate is 10% per year, compounded semiannually, so r = 0.10/2 = 0.05 and n = 2

Marcia wants to have $90,000 in her retirement account

Substituting these values into the formula, we get:

$90,000 = PMT x [(1 + 0.05/2)^(2*15) - 1] / (0.05/2)

Simplifying the formula, we get:

PMT = $90,000 / [(1.025)^30 - 1] / 0.025

PMT = $90,000 / 19.7588

PMT = $4,553.39 (rounded to the nearest cent)

Therefore, Marcia needs to invest $4,553.39 today in order to meet her retirement goal of $90,000, assuming an interest rate of 10% per year, compounded semiannually.

Answer:

[tex]a = \frac{20}{3}[/tex]

Step-by-step explanation:

Dude he played for 1/2 of the game half of 60 is 30.

50%.

I'm I missing something?

After 6 minutes, 266 grams of a radioactive form of krypton will still be present.

"A method to find a single unit value from a multiple unit value and to find a multiple unit value from a single unit value."

We always count the unit or amount value first and then calculate the more or less amount value.

For this reason, this procedure is called a unified procedure.

Many set values ​​are found by multiplying the set value by the number of sets.

A set value is obtained by dividing many set values ​​by the number of sets.

Since, its half-life is 3 minutes.

Hence, The amount of a radioactive kind of krypton after 6 minutes is,

⇒ 984 / 4

⇒ 246 grams

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The function represented by the table is linear, as it has a constant rate of change and is represented by a straight line.

Function in mathematics is a relation between two sets, where one set is the input and the other set is the output. Functions are an important tool in mathematics and can be used to describe and model real-world phenomena. Functions take inputs, manipulate them and produce outputs. They can be used to represent relationships between two or more variables, or to represent a complex process. Functions allow us to break down complex problems into smaller, more manageable pieces and to study how changes in one variable affect other variables.

The function represented by the table is linear. It can be determined by the fact that the y-values change by the same amount every time the x-values increase by one unit. In this case, the y-values decrease by 2 each time the x-values increase by one unit. This is an example of a linear function.

Linear functions have the shape of a straight line and are characterized by having a constant rate of change. The constant rate of change is represented by the slope of the line, which in this case is -2. This means that for every one unit increase in the x-values, the y-values decrease by two.

A quadratic function is the opposite of a linear function, as it has a rate of change that is not constant. Quadratic functions are characterized by their parabolic shape and their rate of change increases as x-values increase. Exponential functions are characterized by their curved shape and increase exponentially as x-values increase.

In conclusion, the function represented by the table is linear, as it has a constant rate of change and is represented by a straight line.

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The peak of the histogram appears at age 19, and its shape is approximately symmetric. This indicates that the students' ages are evenly distributed, with a mean age of around 19.

Probability is a gauge of how likely an occurrence is to take place. It is expressed as a number between 0 and 1, with 0 designating an impossibility and 1 designating a certainty for the occurrence.

For instance, if you flip an impartial coin, you could get either heads or tails. Because there is an equal possibility that the coin will land on its head or tails, the probability of getting heads on a single toss is [tex]0.5[/tex] , or 50%.

Given

c) To determine the likelihood that a pupil chosen at random is under 20 years old, we must add the probabilities of the top two bars in the histogram. Following are the results: P(age 20) = P(age = 17) + P(age = 18)  [tex]= 0.05 + 0.15 = 0.2[/tex]

The likelihood that a pupil chosen at random is under  [tex]20[/tex] years old is therefore  [tex]0.2, or 20%[/tex] .

d) We must add the probabilities of the bars between the ages of 19 and 21, inclusive, in order to determine the likelihood that an arbitrarily chosen student is older than 18 but not older than 21. P(18 age 21) equals P(age = 19) + P(age = 20) + P(age = 21)  [tex]= 0.35 + 0.25 + 0.1 = 0.7[/tex] .

Therefore,  [tex]0.7, or 70%[/tex] , of an arbitrarily chosen student having an age that is greater than 18 but not greater than [tex]21[/tex]  .

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The value of x in the given equation is 15/23.

Two expressions are combined in an equation by an equal symbol ("="). The "left-hand side" and "right-hand side" of the equation are the two expressions on either side of the equals symbol. Typically, we consider an equation's right side to be negative. Since we can balance this by deducting the right-side expression from both sides' expressions, this won't decrease the generality.

in the given equation, 12x+6y=12y=x+15

12x+6y=12y

2x+y=2y

y=2x

now we have,

12y=x+15

12(2x)=x+15

24x=x+15

x=15/23

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The likelihood that one or more of the dice will land on an even number is 1296.

The likelihood of an event is quantified by its probability, which is a number. It is stated as a number between 0 and 1, or in percentage form, as a range between 0% and 100%. The likelihood of an event increasing with probability of occurrence.

Four 6-sided dice are rolled what is the probability that at least two dice show least 2 die the same.

For 2 of the same: 5×5×642) =900

For 3 of the same: 5×643) =120

For 4 of the same: 644) =6

Combined: 900+120+6=1026

Total possibilities: 64=1296

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The probability that at least one of the dice lands on an even number is 15/16 or approximately 0.938.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

We can solve this problem by finding the probability that all four dice land on odd numbers and then subtracting this probability from 1 to get the probability that at least one of the dice lands on an even number.

The probability that one dice lands on an odd number is 3/6 = 1/2, and the probability that all four dice land on odd numbers is:

(1/2) × (1/2) × (1/2) × (1/2) = 1/16

Therefore, the probability that at least one of the dice lands on an even number is:

1 - 1/16 = 15/16

So the probability that at least one of the dice lands on an even number is 15/16 or approximately 0.938.

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The area of a sector of a circle is equal to the central angle in radians multiplied by the square of the radius.

Therefore, we can use this formula to solve for the radius:

Area = (Central Angle) × (Radius)²

Therefore, we can rearrange this formula to solve for the radius:

Radius = √(Area / Central Angle)

Plugging in the values given, we get:

Radius = √(15 cm² / 5 rad)

Simplifying, we get:

Radius = 3 cm

Therefore, the radius of the circle is 3 cm, rounded to the nearest tenth.

By using trigonometry, we find that Traveling directly from New York City to Buffalo would save approximately 50 miles.

To find the angle at which the engineer should build the highway, we can use trigonometry. Let's call the distance between New York City and Buffalo "d". Then, the distance traveled if going through Albany would be 170 + 280 = 450 miles.

We can use the cosine function to find the angle x:

cos(x) = adjacent/hypotenuse = d/450

x = cos^(-1)(d/450)

To find the value of x, we need to know the value of d. If we assume that the distance between New York City and Buffalo is approximately 400 miles, then:

x = cos^(-1)(400/450) = 23.6 degrees (rounded to the nearest tenth)

Therefore, the engineer would need to build the highway at an angle of approximately 23.6 degrees.

To find how many miles would be saved by traveling directly from New York City to Buffalo rather than going through Albany, we can subtract the distances:

450 - d = 450 - 400 = 50 miles (rounded to the nearest mile)

Therefore, traveling directly would save approximately 50 miles.

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The radius of the circle defined by the equation (x-x₁)² + (y-y₁)² is given by the formula √((x - x₁)² + (y - y₁)²).

You have been asked to find the radius of a circle that is defined by the equation (x-₁)² + (y-₁)². This equation represents all the points that are a certain distance away from the point (₁,₁).

To find the radius, we need to determine the distance from the center (₁,₁) to any point on the circle. Since the circle is defined by the equation (x- x₁)² + (y- y₁)², we can use the distance formula to find the radius.

The distance formula states that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:

distance = √((x₂ - x₁)² + (y₂ - y₁)²)

In this case, we want to find the distance from the center (₁,₁) to a point on the circle (x,y). Using the distance formula, we can write:

radius = √((x - x₁)² + (y - y₁)²)

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The quadratic function's zeros are therefore  [tex]x = 1[/tex] and [tex]x = 0.2[/tex] . A degree two polynomial in one or so more variables that is a quadratic function.

Three points are used to determine a quadratic function, which has the form [tex]f(x) = ax2 Plus bx + c.[/tex]

[tex]Sqrt(b2 - 4ac) = [-b sqrt(b)][/tex] Where the quadratic function's coefficients are a, b, and c.

Here,  [tex]a = 20[/tex] , [tex]b = -19[/tex] , & [tex]c = 3[/tex]  . We obtain the quadratic formula by substituting these values: [tex]x = [-(-19) sqrt((-19)2 - 4(20)(3)] / 2(20) (20)[/tex]

When we condense this phrase, we get:

[tex]x = [19 +/- sqrt(361 - 240)] / 40 x = [19 +/- sqrt(121)] / 40\sx = [19 ± 11] / 40[/tex]

Therefore, The zeros of a quadratic equation   [tex]f(x) = 20x2 - 19x + 3[/tex] are as follows: [tex]x = (19 Plus 11) / 40 = 1 and x = (19 − 11) / 40 = 0.2.[/tex]

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Answer:

x = [tex]\frac{7\pi }{6}[/tex], [tex]\frac{11\pi }{6}[/tex]  or x = 210°, 330°

Step-by-step explanation:

4sin(x) + 2 = 0

4sin(x) = -2

sin(x) = -1/2

x = [tex]\frac{7\pi }{6}[/tex], [tex]\frac{11\pi }{6}[/tex]