(essay making) how important factoring polynomials in our lives

The quadratic function that fits the given points is: f(x) = (4/3)x² - (10/3)x + 1

By using the simplification formula

f(x) = ax² + bx + c

where a, b, and c are constants

a(-1)² + b(-1) + c = 1/3

a(0)² + b(0) + c = 1

a(1)² + b(1) + c = 3

a(2)² + b(2) + c = 9

Simplifying each

a - b + c = 1/3

c = 1

a + b + c = 3

4a + 2b + c = 9

We can solve this using any method like substitution, elimination

a - b + c = 1/3

a + b + c = 3

2a + 2c = 9/3

Adding the first two equations 2a + 2c = 10/3

Subtracting the third equation

b = 5a/3 - 2

a - (5a/3 - 2) + 1 = 1/3

a = 4/3

Finally, we can substitute a = 4/3 and b = 5a/3 - 2, and c = 1 into the standard form of the quadratic equation to get: f(x) = (4/3)x² - (10/3)x + 1.

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Answer:

A: [tex]2x^3+5x^2-8x[/tex]

Step-by-step explanation:

In order to get the answer to this, you have to combine like terms to simplify the answer.

Go through and organize the x's by the size of the exponents. (this step isn't necessary but it can help you visualize it if you are having trouble with that)

[tex]x^3+x^3+2x^2+3x^2-7x-x[/tex]

When the variables are raised to the same degree, the coefficients can be added together.

[tex]2x^3+5x^2-8x[/tex]

so we're assuming there are 360 days in a year, so 83 days is really just 83/360 of a year, so

[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$586.21\\ r=rate\to 6.3\%\to \frac{6.3}{100}\dotfill &0.063\\ t=years\dotfill &\frac{83}{360} \end{cases} \\\\\\ I = (586.21)(0.063)(\frac{83}{360}) \implies I \approx 8.51[/tex]

Answer:

Step-by-step explanation:

Solution:

   [tex](x-2)^2+(y+3)^2=25[/tex]

Equation of general circle:

    [tex](x-a)^2+(y-b)^2=r^2[/tex]      where (a,b) is circle center and r is the radius.

The solution to the problem is that the number "x" must be greater than or equal to -12.

Let's define the variable as "x," representing the unknown number. According to the problem statement, one fourth of the number is no less than -3.

Mathematically, we can express this statement as:

x/4 ≥ -3

To solve for "x," we can start by multiplying both sides of the inequality by 4 to eliminate the fraction:

x ≥ -3*4

x ≥ -12

Therefore, the solution to the problem is that the number "x" must be greater than or equal to -12.

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Answer:

Step-by-step explanation:

500/0.4 = 1250

D. Angle Addition Postulate is necessary when proving that the angles of a parallelogram are congruent.

What is parallelogram?

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure.

To prove that opposite angles of a parallelogram are congruent, we need to use the properties of parallelograms, such as opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

We do not need the Segment Addition Postulate, as it is used to find a length of a segment, not to prove the congruence of angles.

We also do not need to use the Opposite sides are perpendicular property, as this is only true for a rectangle or a rhombus, but not necessarily for a parallelogram.

Similarly, the Angle Addition Postulate is used to find the measure of an angle, not to prove that angles are congruent.

Therefore, the only necessary property to prove that opposite angles of a parallelogram are congruent is that opposite angles are congruent, which is a property of parallelograms.

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The total surface area of the given hemispherical scoop is: 30.41 cm²

The formula for the total surface area of a hemisphere is:

TSA = 3πr² square units

Where:

π is a constant whose value is equal to 3.14 approximately.

r is the radius of the hemisphere.

Since the steel is 0.2cm thick and the outside of the scoop has a radius of 2cm, then we can say that:

TSA = 3π(2 + 0.2)²

= 3π(2.2)²

= 30.41 cm²

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Answer:

50 spherical candles can be made with 152,604 cm³ of wax.

Step-by-step explanation:

Since the wax candles are in the shape of a solid sphere, we can calculate the volume of one candle by using the volume of a sphere formula.

[tex]\boxed{\begin{minipage}{4 cm}\underline{Volume of a sphere}\\\\$V=\dfrac{4}{3} \pi r^3$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

The diameter of a sphere is twice its radius.

Therefore, if the diameter of the spherical candle is 18 cm, its radius is:

[tex]\implies r=\dfrac{d}{2}=\dfrac{18}{2}=9\; \sf cm[/tex]

Substitute r = 9 and π = 3.14 into the formula to calculate the volume of one spherical candle.

[tex]\begin{aligned}\implies \textsf{Volume of one candle}&=\sf \dfrac{4}{3} \cdot 3.14 \cdot (9\;cm)^3\\\\&=\sf \dfrac{4}{3} \cdot 3.14 \cdot 729\;cm^3\\\\&=\sf 3052.08\; \sf cm^3\end{aligned}[/tex]

Given the company has a total of 152,604 cm³ of wax, to calculate how many candles can be made, divide the total amount of available wax by the wax needed to make one candle.

[tex]\begin{aligned}\textsf{Total number of candles}&=\sf \dfrac{152604\; cm^3}{3052.08 \;cm^3}\\\\&=\sf 50\end{aligned}[/tex]

Therefore, 50 spherical candles can be made with 152,604 cm³ of wax.

Answer: Let's start by using the identity:

tan(arctan(x)) = x

to simplify the expression inside the sine function. So, we have:

arctan(u) / sqrt(3) = tan(arctan(u) / sqrt(3))

Now, using the trigonometric identity:

tan(x/y) = sin(x) / (cos(y) + sin(y))

with x = arctan(u) and y = sqrt(3), we get:

tan(arctan(u) / sqrt(3)) = sin(arctan(u)) / (cos(sqrt(3)) + sin(sqrt(3)))

Simplifying further, we know that:

sin(arctan(u)) = u / sqrt(1 + u^2)

and

cos(sqrt(3)) + sin(sqrt(3)) = 2cos(sqrt(3) - pi/4)

So, the expression becomes:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2cos(sqrt(3) - pi/4)]

Simplifying the denominator, we have:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / (2(cos(sqrt(3))cos(pi/4) + sin(sqrt(3))sin(pi/4)))

Using the values for cosine and sine of pi/4, we get:

cos(pi/4) = sin(pi/4) = 1/sqrt(2)

So, we have:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2(sqrt(3)/2 + 1/2)]

Simplifying further:

sin(arctan(u) / sqrt(3)) = u / (sqrt(1 + u^2) * (sqrt(3) + 1))

Therefore, the algebraic expression for sin(arctan(u) / sqrt(3)) is:

u / (sqrt(1 + u^2) * (sqrt(3) + 1))

Step-by-step explanation:

If the roots of a quadratic equation are given, we can write the equation in factored form as [tex](x - r1)(x - r2) = 0[/tex] , where r1 and r2 are the roots.the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

A quadratic equation can be written in the form:

[tex]ax^2 + bx + c = 0[/tex]

where a, b, and c are constants. Since the roots of the equation are 5 and 2, we can write:

[tex](x - 5)(x - 2) = 0[/tex]

Expanding this equation gives:

[tex]x^2 - 7x + 10 = 0[/tex]

To make the leading coefficient of this equation 4, we can multiply both sides by 4/1, which gives:

[tex]4x^2 - 28x + 40 = 0[/tex]

Therefore, the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

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Answer:

581,58 in^3

Step-by-step explanation:

Given:

Cylinder shaped cups

d (diameter) = 4,2 in

r (radius) = 0,5 × 4,2 = 2,1 in

h (height) = 7 in

π = 3.14

.

First, let's find how much juice can he pour into 1 cup:

.

We need to find the base of the cylinder:

.

[tex]a(base) = \pi {r}^{2} = 3.14 \times( {2.1})^{2} = 13.8474[/tex]

.

Now, we can find the volume of one cup:

V = a (base) × h

[tex]v = 13.8474 \times 7 ≈96.93[/tex]

Multiply this number by 6 and we'll get the answer (since there's 6 cups):

96,93 × 6 = 581,58

Answer:

30

Step-by-step explanation:

its right

order of operations

Answer:

The answer is 30...

Step-by-step explanation:

Apply the rule BODMAS...

3(40)/(4)

120/4

30

The required answer is the speed limit of 90 km/h is equivalent to approximately 56 mph.

To convert the speed limit from kilometers per hour (km/h) to miles per hour (mph), use the conversion factor of 1 km/h = 0.621371 mph. Let's follow these steps:

Step 1: Determine the conversion factor.

Since we know the conversion factor is 1 km/h = 0.621371 mph,  use this to convert the speed limit from km/h to mph.

Step 2: Calculate the speed limit in mph.

The given speed limit is 90 km/h. Multiply this value by the conversion factor to obtain the equivalent speed in mph:

90 km/h x 0.621371 mph/km/h = 55.92339 mph

Step 3: Round the result, if necessary.

To provide a practical speed limit, we can round the result to the nearest whole number. Therefore, the speed limit in miles per hour is approximately 56 mph.

Hence, the speed limit of 90 km/h is equivalent to approximately 56 mph.

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Step-by-step explanation:

6.

the figure can be considered as a combination of a 16×8 rectangle and a 10×8 right-angled triangle on the left side.

for both, perimeter and area, we need to find the length of the missing third side of the right-angled triangle, as this is a part of the long baseline of the overall figure.

as it is a right-angled triangle, we can use Pythagoras :

c² = a² + b²

"c" being the Hypotenuse (the side opposite of the 90° angle, in our case 10 ft). "a" and "b" being the legs (in our case 8 ft and unknown).

10² = 8² + (leg2)²

100 = 64 + (leg2)²

36 = (leg2)²

leg2 = 6 ft

that means the bottom baseline is 16 + 6 = 22 ft long.

a.

Perimeter = 10 + 16 + 8 + 22 = 56 ft

b.

Area is the sum of the area of the rectangle and the area of the triangle.

area rectangle = 16×8 = 128 ft²

area triangle (in a right-angled triangle the legs can be considered baseline and height) = 8×6/2 = 24 ft²

total Area = 128 + 24 = 152 ft²

7.

it was important that the bottom left angle of the quadrilateral was indicated as right angle (90°). otherwise this would not be solvable.

but so we know, it is actually a rectangle.

that means all corner angles are 90°.

therefore, the angle AMT = 90 - 20 = 70°.

the diagonals split these 90° angles into 2 parts that are equal in both corners of the diagonal, they are just up-down mirrored.

a.

so, the angle HTM = AMT = 70°.

b.

MEA is an isoceles triangle.

so, the angles AME and EAM are equal.

the angle AME = AMT = EAM = 70°.

the sum of all angles in a triangle is always 180°.

therefore,

the angle MEA = 180 - 70 - 70 = 40°

c.

both diagonals are equally long, and they intersect each other at their corresponding midpoints.

so, when AE = 15 cm, then AH = 2×15 = 30 cm.

TM = AH = 30 cm.

Answer:

The answer to your problem is, 75

Step-by-step explanation:

Yellow = [tex]\frac{1}{2}[/tex] of the marbles

Red = [tex]\frac{1}{8}[/tex] of the marbles

Blue = [tex]1 - \frac{1}{2} - \frac{1}{8} = \frac{1}{2} - \frac{1}{8}[/tex]

= [tex]\frac{1*4}{2*4} - \frac{1}{8}[/tex]

= [tex]\frac{1}{2} or \frac{4}{8} - \frac{1}{8}[/tex]

= [tex]\frac{3}{8}[/tex]

[tex]\frac{3}{8}[/tex] of the marbles are represented as blue.

200 x [tex]\frac{3}{8}[/tex] = 25 x 3 = 75

Thus the answer to your problem is, 75

Answer:

13.2 miles

Step-by-step explanation:

We can use the following conversion factors:

1 mile = 5,280 feet

Using this conversion, we can divide the height of the volcano in feet by 5,280 to get the height in miles:

69,657.652 ft ÷ 5,280 ft/mi ≈ 13.2 mi

Therefore, the height of the volcano on the recently discovered planet is approximately 13.2 miles.

Hopes this helps

well, let's keep in mind that since a year has 12 months, then 17 months is really 17/12 of a year, so

[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$1400\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ t=years\dotfill &\frac{17}{12} \end{cases} \\\\\\ I = (1400)(0.05)(\frac{17}{12}) \implies I \approx 99.17[/tex]

It can be expressed in interval notation as:[0, 12]

The practical domain for the function g(x) is [0,12], as this is the time during which the football is in the air, from the time it is thrown until it hits the ground.

The practical domain for the function g(x) would be the time interval during which the football is in the air, since it only makes sense to talk about the height of the football while it is airborne.

From the problem statement, we know that the football is thrown in the air at time x = 0, reaches its maximum height of 28 feet at time x = 6, and hits the ground at time x = 12. Therefore, the practical domain for the function g(x) is:
0 <= x <= 12
This means that the function g(x) is defined and meaningful for any value of x between 0 and 12, inclusive. Beyond this domain, the function does not have a practical interpretation because the football is either not yet thrown or has already hit the ground.

The function g(x) is the height of a football x seconds after it is thrown in the air. The football reaches its maximum height of 30 feet in 4 seconds, and hits the ground at 10 seconds.
What is the practical domain for the function f(x)
Write your answer in interval notation.

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Answer:

a) 4.46 miles

b) 3 miles

Step-by-step explanation:

Law of Sines:

[tex]\dfrac{\text{a}}{\text{sin(A)}} =\dfrac{\text{b}}{\text{sin(B)}}[/tex]

a) The distance of the plane from point A

The angle of depression corresponds to the congruent angle of elevation therefore, 180 - 28 - 52 = 100°

[tex]\dfrac{\text{6.7}}{\text{sin(100)}} =\dfrac{\text{b}}{\text{sin(41)}}[/tex]

[tex]\text{b}=\dfrac{6.7\text{sin}(41)}{\text{sin}(100)}[/tex]

[tex]\text{b}=4.46 \ \text{miles}[/tex]

b) Elevation of the plane

[tex]\text{sin}=\dfrac{\text{opposite}}{\text{hypotenuse}}[/tex]

hypotenuse is 4.46 and opposite is the elevation(h) to be found

[tex]\text{sin}(38)=\dfrac{\text{h}}{4.46}[/tex]

[tex]\text{h}=\text{sin}(38)4.46[/tex]

[tex]\text{h}=3[/tex]

Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).

A polyhedron with two triangular sides and three rectangles sides is referred to as a triangular prism. It is a three-dimensional shape with two base faces, three side faces, and connections between them at the edges.

Given :

We must calculate the area of each face of the triangular prism and put them together to determine its surface area.

The areas of the triangular faces are equal, so we may calculate one of their areas and multiply it by two:

One triangular face's area is equal to (1/2) the sum of its base and height, or (1/2) 6 x 8 x 6, or 24 square units.

Both triangular faces' surface area is 2 x 24 or 48 square units.

Finding the area of the rectangular faces is now necessary:

One rectangular face's area is given by length x breadth (10 x 6) = 60 square units.

120 square units are the area of both rectangular faces or 2 by 60.

Hence, the triangular prism's total surface area is:

Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).

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The formula used to calculate the surface area of a triangular prism is:  

S = bh + 2ah  

Where S = surface area, b = length or side of the triangle, and h = height of the triangle.  

For right triangles, the height can be calculated as:  

c = a^(2) \+ b^(2)  

c = the hypotenuse  

a and b = the two sides of the triangle

(A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

What is the rate of change?

The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.

A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.

Using the formula for an annual rate of change (r):

final value = initial value * [tex](1 - r)^t[/tex]

where t is the number of years and r is the annual rate of change expressed as a decimal.

Substituting the given values, we get:

$15,000 = $44,000 * (1 - r)¹⁴

Solving for r, we get:

r = 0.0804

So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.

B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:

r = 0.0804 * 100% = 8.04%

C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.

Substituting the known values, we get:

value in 2009 = $15,000 * (1 - 0.0804)³

value in 2009 = $11,628.40

Rounding to the nearest $50, we get:

value in 2009 = $11,650

Hence, (A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

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The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

Probability refers to the likelihood that an occurrence will occur.

In actual life, we frequently have to make predictions about the future. We might or might not be conscious of how an event will turn out.

When this occurs, we proclaim that there is a possibility that the event will occur. In conclusion, probability has a broad range of amazing uses in both business and this rapidly developing area of artificial intelligence.

Simply dividing the favorable number of possibilities by the total number of possible outcomes using the probability formula will yield the chance of an event.

According to our question-

Total number of books that arrived that day = 23 + 41 = 64.

Let E be the event of arriving at a non-fictional book.

The event of arriving at a non-fictional book is n(E) = 23.

Total sample space, n(S) = 64.

The probability of a non-fictional book to arrive is = n(E) / n(S)

= 23 / 64 = 0.359375

= 0.359375 × 100

= 35.9375 ≈ 36.

Hence, The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

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Answer:

Without knowing the exact length of Ted's pencil, we cannot give an exact decimal representation of its length. However, we do know that Ted's pencil is longer than Jan's pencil, which is 8.5 cm long.

If we assume that Ted's pencil is one centimeter longer than Jan's pencil, then its length would be 9.5 cm. In decimal form, this would be written as 9.5.

If we assume that Ted's pencil is two centimeters longer than Jan's pencil, then its length would be 10.5 cm. In decimal form, this would be written as 10.5.

So, the decimal that could represent the length of Ted's pencil depends on how much longer it is than Jan's pencil.

The side of the given square is 36W^2 + 12W +1 of is 6W +1 and 81W^2 -72W + 16 is (9W-4).

The area is calculated by multiplying the length of a shape by its width.

and the unit of the square is a square unit.

Given Area of the square :

1) [tex]36W^{2}+12W+1[/tex]

Area of square = [tex]36W^{2}+12W+1[/tex]

[tex]side^{2}[/tex] = [tex]36W^{2} + 6W +6W +1\\6W (6W +1) +1 (6W +1)\\(6W +1)(6W+1)\\(6W +1 )^{2} \\side^{2} = (6W + 1)^{2} \\side = 6W + 1[/tex]

[tex]Area of the square = 81W^{2} - 72W + 16\\(side)^{2} = (9W-4)(9W-4)\\(side)^{2} = (9W-4)^{2} \\side = (9W-4)[/tex]

Therefore the side of the square is 6W+1 and 9x-4.

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Answer:

area=28cm^2

Step-by-step explanation:

4×5=20

20÷2=10 area of the bigger triangle

4×2=8

8÷2=4 area of the smaller triangle

10+10=20 for area of the the 2 bigger triangles

4+4=8 for the area of the 2 smaller triangles

area of kite= 20+8= 28

Answer:

the answer is A) -2x² + 4x + 30

Step-by-step explanation:

To find the equation of the quadratic function f(x), we can use the standard form of a quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants.

We can plug in the values of x and f(x) from the table to get three equations:

a(-1)^2 + b(-1) + c = 24

a(0)^2 + b(0) + c = 30

a(1)^2 + b(1) + c = 32

Simplifying each equation, we get:

a - b + c = 24

c = 30

a + b + c = 32

We can substitute c = 30 into the first and third equations to get:

a - b + 30 = 24

a + b + 30 = 32

Simplifying these equations, we get:

a - b = -6

a + b = 2

Adding these two equations, we get:

2a = -4

Dividing by 2, we get:

a = -2

Substituting a = -2 into one of the equations above, we get:

-2 - b = -6

Solving for b, we get:

b = 4

Therefore, the equation that represents f(x) is:

f(x) = -2x^2 + 4x + 30

So the answer is A) -2x² + 4x + 30