find all values of c such that the domain of the following function is the set of all real numbers
f(x) = (x + 7) / (x^2 + 2cx + 14)

Answer:  see proof below

Step-by-step explanation:

Use the following Power Reducing Identities:

[tex]\sin^3\theta= \dfrac{1+\cos 2\theta}{2}\\\\\\\cos^2\theta=\dfrac{3\sin \theta-\sin 3\theta}{4}[/tex]

Use the following Product to Sum Identity:

[tex]\sin A \cdot \cos B = \dfrac{1}{2}\bigg[\sin (A + B) + \sin (A - B)\bigg][/tex]

Proof LHS → RHS

LHS:                           cos² Ф · sin³ Ф

[tex]\text{Power Reducing:}\qquad \bigg(\dfrac{1+\cos 2\theta}{2}\bigg)\bigg(\dfrac{3\sin \theta-\sin 3\theta}{4}\bigg)[/tex]

[tex]\text{Expand:}\qquad \qquad \dfrac{1}{8}(3\sin \theta -\sin 3\theta +3\sin \theta\cdot cos 2\theta-\sin 3\theta \cdot \cos 2\theta)[/tex]

[tex]\text{Product to Sum:}\\ \dfrac{1}{8}\bigg[3\sin \theta -\sin 3\theta +\dfrac{3}{2}\bigg(\sin (\theta +2\theta) +\sin(\theta -2\theta)\bigg)-\dfrac{1}{2}\bigg(\sin (3\theta+2\theta) +\sin (3\theta-2\theta)\bigg)\bigg][/tex]

[tex]\text{Simplify:}\quad \dfrac{1}{8}\bigg[3\sin \theta -\sin 3\theta +\dfrac{3}{2}\bigg(\sin 3\theta +\sin(-\theta)\bigg)-\dfrac{1}{2}\bigg(\sin 5\theta +\sin \theta\bigg)\bigg][/tex]

[tex]\text{Cofunction:}\quad \dfrac{1}{8}\bigg[3\sin \theta -\sin 3\theta +\dfrac{3}{2}\bigg(\sin 3\theta -\sin(\theta)\bigg)-\dfrac{1}{2}\bigg(\sin 5\theta +\sin \theta\bigg)\bigg][/tex]

[tex]\text{Simplify:}\qquad \dfrac{1}{16}\bigg(6\sin \theta -2\sin 3\theta +3\sin 3\theta -3\sin(\theta)-2\sin 5\theta -2\sin \theta\bigg)\\\\\\.\qquad \qquad =\dfrac{1}{16}\bigg(2\sin \theta + \sin 3\theta -\sin 5\theta\bigg)[/tex]

[tex]\text{LHS=RHS:}\quad \dfrac{1}{16}\bigg(2\sin \theta + \sin 3\theta -\sin 5\theta\bigg)=\dfrac{1}{16}\bigg(2\sin \theta + \sin 3\theta -\sin 5\theta\bigg)\quad \checkmark[/tex]

Answer:

This answer assumes the vertices of the polygon are at A(1, 2), B(2, 3), C(3, 2), and D(2, 1).

Step-by-step explanation:

A (1, 2)  Ax (1, −2)  Ay (−1, 2)

B (2, 3) Bx (2, −3)  By (−2, 3)

C (3, 2) Cx (3, −2)  Cy (−3, 2)

D (2, 1)  Dx (2, −1)  Dy (−2, 1)

Answer:

Vertex Coordinate  Vertex Coordinate  Vertex Coordinate

A (1, 2)  Ax (1, −2)  Ay (−1, 2)

B (2, 3)  Bx (2, −3)  By (−2, 3)

C (3, 2)  Cx (3, −2)  Cy (−3, 2)

D (2, 1)  Dx (2, −1)  Dy (−2, 1)

Step-by-step explanation:

Plato Sample Anwser

Step-by-step explanation:

i hope it helped you a lot. thankyou

Answer:

y=8

Step-by-step explanation:

Horizontal equations are of the form

y =

The y coordinate is 8

y=8

Answer:

Negative

Step-by-step explanation:

A negative plus a negative (negative minus a positive) will give you a negative. -1.69 + (-1.69) = -3.38

-3.38 is a negative, therefore the answer is negative.

Answer:

The range of the function are the values of height which are from 3, 4, 5, ..., 50

Step-by-step explanation:

The given parameters are;

The height at which the ball is caught = 5 feet

The height at which the ball was hit = 3 feet

The maximum height reached by the ball = 50 feet

The height of the ball given as a function of time is f(t) s = u·t - 1/2·g·t²

Where;

s = 50 feet

g = 9.81 m/s²

Therefore, we have;

50 = u·t - 1/2 × 9.81 × t²

50 = u·t - 4.905 × t²

Therefore, the range are obtainable values for f(t) which range from 3 to 50.

Yes

If the rectangle has sides of Square root 8 and square root 2 which are two irrational numbers but when multiplied it gives an area of 4 square units which is a rational number.

No, it is not possible for a rectangle to have rational side lengths and an irrational area, because two rational numbers are always rational.

Given that,
To figure out, is it possible for a rectangle to have rational side lengths and an irrational area.

Rational numbers can be structured in the form of the fraction of integers. Eg- 5/6, 2/3, etc.

Here,
in a rectangle, two sides have rational length and the area of the rectangle is a product of the sides of the rectangle.
Since there is a property of the rational numbers that, when two rational numbers are multiplied together the product of the rational number is always rational.

So the area of the rectangle also would be rational.

Thus,  it is not possible for a rectangle to have rational side lengths and an irrational area.

Learn more about Rational numbers here: https://brainly.com/question/17450097

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Answer:

- 11 or - 15

Step-by-step explanation:

Factorise

x² - 13x + 42

Consider the factors of the constant term (+ 42) which sum to give the coefficient of the x- term (- 13)

The factors are - 6 and - 7, since

- 6 × - 7 = + 42 and - 6 - 7 = - 13, thus

x² - 13x + 42 = (x - 6)(x - 7)

compare to (x + A)(x + B)

A = - 6, B = - 7

thus 3A - B = 3(- 6) - (- 7) = - 18 + 7 = - 11

The factors may also be expressed as (x - 7)(x - 6)

with A = - 7 and B = - 6, then

3A - B = 3(- 7) - (- 6) = - 21 + 6 = - 15

Answer:

X is the real part

Step-by-step explanation:

There's a bit of lack of context but I think the question is just asking what variables to put in for Lucy and Howard. There seems to be no need to answer any math problem.

I suggest using the variables x and y or l for Lucy and h for Howard. Remember to use lowercase letters.

Answer:

1) x = 0

2) x = 2

Step-by-step explanation:

1)

12 = -4(-6x - 3)

First, we distribute.

12 = 24x + 12

Then, we combine like terms.

12 - 12 = 24x

0 = 24x

Finally, we divide 24 on both sides.

We got 0 = x, or x = 0

-

2)

-20 = -4x - 6x

First, we simplify.

-20 = -10x

Then, we divide -10 on both sides.

We got 2 = x, or x = 2

-

Hope this helps!

the answer is 30( -3 3/5*(-8 1/3

Answer:

the second choice

Step-by-step explanation:

[tex] \dfrac{1}{2}n + 3 < 5 [/tex]

Subtract 3 from both sides.

[tex] \dfrac{1}{2}n < 2 [/tex]

Multiply both sides by 2.

[tex] n < 4 [/tex]

To show n < 4 on a number line, you put an open circle on 4, and you color the line to the left of 4.

Answer: the second choice

Answer:

Step-by-step explanation:

7y+5+3y+5=70

10y+10=70

10y=60

y=6

7(6)+5= 42+5= 47

3(6)+5= 18 + 5 =23

47+23= 70

Answer:  Answer is in the steps

Step-by-step explanation:

To determine if their are congruent ,find the distance between the points.

In AB  it has the points (-6,-4) and (-1,8)  

Now to find the distance you will have to find the difference between the x and y values then square them to add them and find the square root.

-4 - 8 = -12

-6 - (-1) = -5

12^2 + 5^2 =d^2  where d is the distance so solve for d

144 + 25 = d^2

169 =d^2

d = [tex]\sqrt{169}[/tex]  

d= 13  

So we now the distance of AB is 13  

now we need to find the distance between CD

(2,5)  (14,0)  using these points follow the same steps in the first step to find the distance.

5-0= 5

2 -14 = -12

5^2 + 12^2 = d^2

25 + 144 = d^2

169 =d^2

d = [tex]\sqrt{169}[/tex]  

d= 13

So using the information we can say the segments are congruent because they have the same distances .

Answer:

Volume = 36.4 m^3 or 9616.6 gallon

weight = 18025.47 N or 80213.33 lbs

Step-by-step explanation:

The complete question is

A small family home in Tucson, Arizona, has a rooftop area of 1967 square feet, and it is possible to capture rain falling on about 56% of the roof. A typical annual rainfall is about 14 inches. If the family wanted to install a tank to capture the rain for an entire year, without using any of it, what would be the required volume of the tank in m3 and in gallons? How much would the water weigh when the tank was full (in N and in lbf)?

The area of the roof = 1967 ft^2

Area of water to be possibly captured = 56% of 1967 ft^2 = 0.56 x 1967 = 1101.52 ft^2

Rainfall depth = 14 inches = 14/12 ft = 1.167 ft

Volume of water = (area of water) x (depth of water) = 1101.52 ft^2 x 1.167 ft = 1285.47 ft^3

35.315 ft^3 = 1 m^3

1285.47 ft^3 = [tex]x[/tex] m^3

==> 1285.47/35.315 = 36.4 m^3

1 ft^3 = 7.481 gallon

1285.47 ft^3 = [tex]x[/tex] gallon

==> 1285.47 x 7.481 = 9616.6 gallon

Weight of water W = ρv

where ρ is the density of water = 62.4 lbs/ft^3

v is the volume of water = 1285.47 ft^3

substituting values,

W = 62.4 x 1285.47 = 80213.33 lbs

but,

1 lb = 4.45 N

therefore,

80213.33 lbs = 80213.33/4.45 = 18025.47 N

━━━━━━━━━━━━━━━ ♡ ━━━━━━━━━━━━━━━  

If Ali has x cards, then Belinda has 2x cards.

If Ali has x cards, then Charlie has x+5 cards.

2x (Belinda) + x+5 (Charlie) + x (Ali) = 33 (total)

2x + x+5 + x = 33

Now simplify the equation... x+x+5 can be simplified as 2x+5.

2x+5 plus 2x can become 4x+5.

So their total number is 4x+5.

So 4x+5=33.

You want to isolate the variables and numbers.

To remove the 5 from the left side, subtract 5 from both sides.

4x+5-5=4x - 33-5=28

So the equation now is 4x=28

To remove the 4, divide each side by 4.

4x÷4=x - 28÷4=7

So x equals 7, Ali has 7 cards.

━━━━━━━━━━━━━━━ ♡ ━━━━━━━━━━━━━━━  

Answer:

It's the third option

Step-by-step explanation:

the domain is all the x values

Answer:

Step-by-step explanation:

Play 1 they lose 4y

-4

Play 2 they lose 13y

-13

Total so far we have

-17

play three they gain 2y

-17+2=-15

Over the course of three plays the team loses 15 Yards.

[tex]9=7d+8(9-2d)\\9=7d+72-16d\\9=-9d+72\\-72+9=-9d\\-63=-9d\\d=7[/tex]

Also for the 8(9-2d), simplify by multiplying the 8 into the (9-2d) as we get (8*9)-(8*2d) = 72-16d

Therefore, the answer is d = 7 aka C.

Answer:

k = 5

Step-by-step explanation:

[tex](x + 2)(x + 3) = {x}^{2} + 5x + 6[/tex]

Now comparing left hand side & right hand side ,

[tex] {x}^{2} + kx + 6 = {x}^{2} + 5x + 6[/tex]

Cancelling x^2 & 6 from both left hand side & right hand side ,

kx = 5x

=> k = 5

Answer:

See Explanation

Step-by-step explanation:

The question is incomplete has the figure of the triangle is not given.

However, the following steps will guide you.

To solve this question;

you need the lengths of the threes sides of the triangular fields

Represent these lengths with a, b and c

Next;

Sum these lengths together;

This gives the required length of fencing

[tex]Length = a + b + c[/tex]

Take for instance;

[tex]a =4;\ b = 3;\ c = 5[/tex]

The required length of fence would be

[tex]Length = 4 + 3 + 5[/tex]

[tex]Length = 12\ units[/tex]

Apply these steps and you'll get your answer

Answer:

u use the closest number to the number u are rounding up to round it up.if it is 0-4 u round it to 0and add to the number while if it is 5-9 u round it up to 1 and add to the number.after all rounding up the number used to round up downwards will be converted to 0s

Answer:

If the ones place has a number less than 4, the tens stay the same with a zero in the ones place. If the ones place has a number more than 5, the tens will have another ten. If the ones place has a number 5 in the ones place, then you round the tens to the next ten.

Examples: 37=40       73=70          65=70

Answer:

Integer

Step-by-step explanation:

The greatest integer function (as known as the floor function), is defined as to be the greatest integer less than or equal to x.

It is the function:

[tex]f(x)=\floor[x][/tex]

For instance, if x is 2, then f(x) is 2. If x is 2.7, then f(x) is still 2. 2 is the greatest integer less than or equal to 2.

You can also see this from the attached graph.

Answer:

15.231

Step-by-step explanation:

A perfect cube is a number that is the cube of an integer. For example, 125 is a perfect cube since 125 = 5 × 5 × 5= [tex]5^{3}[/tex]. Some examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343, ..

16 does not have a perfect cube, but does have a  perfect square 4×4=16

Answer:

The actual distance between Phoenix and Fountain Hills is 7.3 miles

Step-by-step explanation:

We note that the scale factor gives the ratio of the map reading to the actual distance reading

The scale factor of the map is 1 inch = 2.5 miles

The distance between Phoenix and Fountain Hills = 2.92 inches on the map

Let the actual distance between Phoenix and Fountain Hills = D, we have;

D/(2.92 inches) = 2.5 miles/inch

Therefore;

D = 2.92 inches ×  2.5 miles/inch = 7.3 miles

D = 7.3 miles

The actual distance between Phoenix and Fountain Hills = D = 7.3 miles.

Answer:

(0+m)/2. (0+n)/2

Step-by-step explanation:

U sum both x coordinates and divide by two. U do same to the t values.

The formula used to find the midpoint of a segment with endpoints A(0, 0) and B(m, n) is x = m / 2 and y = n / 2.

What is the midpoint on a given line?

If on a straight line we have two endpoints:

- [tex](x_{1}~,y_{1})[/tex]

- [tex](x_{2},~ y_{2} )[/tex]

and the midpoint is given by ( x, y ).

Then to find the midpoint on a straight line we use the formula:

x = ( [tex]x_{2}[/tex] + [tex]x_{1}[/tex] ) / 2

y = ( [tex]y_{2}+y_{1}[/tex] ) / 2

We have,

[tex](x_{1}~,y_{1})[/tex] = ( 0, 0 )

[tex](x_{2},~ y_{2} )[/tex] = ( m, n )

[tex]x_{1}[/tex] = 0

[tex]x_{2}[/tex] = m

[tex]y_{1}[/tex] = 0

[tex]y_{2}[/tex] = n

We will use the formula :

x = ( [tex]x_{2}[/tex] + [tex]x_{1}[/tex] ) / 2

y = ( [tex]y_{2}+y_{1}[/tex] ) / 2

And substitute the values.

x = ( m + 0 ) / 2

y = ( n + 0 ) / 2

x = m / 2

y = n / 2.

The formula used to find the midpoint of a segment with endpoints A(0, 0) and B(m, n) is x = m / 2 and y = n / 2.

Learn more about midpoints here:

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Answer:

13) (-5,1)

Step-by-step explanation: