Simplify (1 − cos x)(1 + cos x). (2 points)

Answer:

Explanation:

1) Make a change of variable:

2) The new equation with u is:

3) Factor the left side of the new equation:

4) Come back to the equation replacing the left side with its factored form and solve:

5) Come back to the original substitution:

If u = - 3 ⇒ xp = - 3 ⇒ x or p is negative and that is against the condition that x and p are both greater than zero, so this solution is discarded.

Then use the second solution:

Solve for p:

Answer:

2

Step-by-step explanation:

because 2*2*2*2is equal to 16

The real fourth roots of 16 are: 2 and -2.

The detailed answer explains how to find the real fourth roots of 16 step by step, providing a clear solution.

The question: Find all real fourth roots of 16.

Step-by-step explanation:

Answer:

answer is  b. 6x3 + 3x + 2

Step-by-step explanation:

Answer:

9

Step-by-step explanation:

Ok... So there are 12 donuts in a dozen. And if she bought 3 dozen, she will have 36 donuts. If she plans to share them with her 3 friends, there will be 4 people total sharing the donuts. Then all you have to do is divide the 36 donuts among the 4 friends. 36/4=9. So they will all get 9 donuts.

3(12)/4=x

36/4=x

9=x

x=9

Answer:

2.5(x + 2y - 1).

Step-by-step explanation:

2.5x + ( 5y) -2.5

2.5x + 5y - 2.5  (we can just remove the parentheses without changing the value of the expression)

Each coefficient is divisible by 2.5 so by the distributive law:

2.5x + 5y - 2.5

= 2.5(x + 2y - 1).

By factorization of 2.5x + ( 5y) -2.5, the result is as follows-

2.5(x + 2y -1) is the expression equivalent to 2.5x + ( 5y) -2.5

What is factorization?

At first it is important to know about algebraic expression.

Algebraic expression consists of numbers and variables connected with addition, subtraction, multiplication and division.

Factorization is the process of breaking down of an algebraic expression into a product of algebraic expressions of smaller degree. Each smaller degree algebraic expressions are called factors.

Here,

2.5x + 5y - 2.5

2.5(x + 2y -1)

An expression that is equivalent to 2.5x + ( 5y) -2.5 is

2.5(x + 2y -1)

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Answer:

y =  -2x^2 -4x +1

Step-by-step explanation:

y-3 = -2 (x+1)^2

Foil (x+1)^2

(x+1)(x+1) = x^2 +x+x+1 = x^2 +2x+1

Substitute this back in

y-3 = -2(x^2 +2x+1)

Distribute

y-3 = -2x^2 -4x -2

Add 3 to each side

y-3+3 = -2x^2 -4x -2+3

y =  -2x^2 -4x +1

Answer:

27 ham sandwiches

Step-by-step explanation:

Let

x ----> the number of turkey sandwiches

y ----> the number of ham sandwiches

we know that

x=y -----> equation A

y=3(x-18) -----> equation B

substitute equation A in equation B

x=3(x-18)

x=3x-54

3x-x=54

2x=54

x=27 turkey sandwiches (initial)

therefore

The number of ham sandwiches is equal to

y=27 ham sandwiches

Note The options of this problem are incorrect, the number of sandwiches of ham must be greater than 18

Answer:

60

Step-by-step explanation:

We are given the triangles are congruent, that means the angles are the same measurement

<A = <X

<B = <Y

<C = <Z

We know A = 50  so X = 50

We know <Y = 70 so < B = 70

The three angles of a triangle add to 180

<A + <B + <C = 180

Substituting into the equation

50 + 70 + <C = 180

Combining like terms

120 + <C =180

Subtracting 120 from each side

120-120 +<C =180-120

<C = 60

Answer:

∠C = 60°

Step-by-step explanation:

Corresponding angles are congruent, thus

∠B = ∠Y = 70°

The sum of the 3 angles in ΔABC = 180°

∠C = 180° - (70 + 50)° = 180° - 120° = 60°

Answer:

The range is all real numbers.

The domain is all reals numbers that are greater than -4.

Step-by-step explanation:

[tex]y=\log_3(x+4)[/tex] only exists when [tex]x+4[/tex] is positive.

You can take the log of a negative or 0 number.

So [tex]x+4>0[/tex] implies [tex]x>-4[/tex].  (I just subtract 4 on both sides.)

So the domain is x>-4. You should see this also when you graph the curve that the curve only exist to the right of -4.

Now the range.  The range is where the curve exist for the y-values.

The equivalent exponent form of [tex]y=\log_3(x+4)[/tex] is [tex]3^{y}=x+4[/tex]

We can solve this for x be subtract 4 on both sides:

[tex]x=3^y-4[/tex]

Now here y can be anything; there are no restrictions on the exponent.

Also if you look at the graph of [tex]y=\log_3(x+4)[/tex] you should see every y getting hit by the curve (look down to up; use the y-axis as a guide).

Let's think about the inverse I found above a little more (I'm going to swap x and y).

[tex]y=3^x-4[/tex].

If we look at the domain and range of this we can just swap it to get the domain and range of [tex]y=\log_3(x+4)[/tex].

[tex]y=3^x-4[/tex] is an exponential function of 3^x that has been moved down 4 units.

The range since it has been moved down 4 units is [tex](-4,\infty)[/tex].

The domain of an exponential function is all real numbers.  There are no restrictions on what you can plug in for x.

So swapping these to find the domain and range of [tex]y=\log_3(x+4)[/tex]:

Domain:  [tex](-4,\infty)[/tex]

Range : [tex](-\infty,\infty)[/tex]

Answer: For Edg is x> -4

And for the second on it is all real numbers

Answer:

Part A)

1) The graph in the attached figure N 1

2) The coordinate rule is (x,y) -----> (x,y+10)

3) The translation of the function up 10 units means that the initial deposit is $60 instead of $50

Part B)

1) The graph in the attached figure N2

2) The coordinate rule is (x,y) -----> (x-10,y)

3) The translation of the function right 10 units means that the initial deposit is equal to $10

Part C)

1) In each translation, the slope is the same (m=5) are parallel lines

2) The vertical translation would be up 40 units

Step-by-step explanation:

we have

[tex]f(x)=5x+50[/tex]

where

f(x) --> represents Jeremy's account balance

x ---> the time in years

Part A)

The translation of the function is up 10 units.

The rule of the translation is equal to

(x,y) -----> (x,y+10)

so

The new function will be

[tex]f(x)=5x+50+10[/tex]

[tex]f(x)=5x+60[/tex]

The graph in the attached figure N 1

The translation of the function up 10 units means that the initial deposit is $60 instead of $50

Part B)

The translation of the function is right 10 units.

The rule of the translation is equal to

(x,y) -----> (x-10,y)

so

we have

[tex]f(x)=5x+60[/tex] ----> function Part A

The new function will be

[tex]f(x)=5(x-10)+60[/tex]

[tex]f(x)=5x+10[/tex]

The graph in the attached figure N 2

The translation of the function right 10 units means that the initial deposit is equal to $10

Part C)

1. Look at the translations, what characteristic of the graph stayed the same in each translation?

In each translation, the slope is the same

The slope m is equal to m=5  

Are parallel lines

2. Look at the original graph and the graph of the translation right 10 units. What vertical translation of the graph in Part B would put the graph back to its original position?

we have

[tex]f(x)=5x+10[/tex]

The vertical translation would be up 40 units

The rule of the translation is equal to

(x,y) -----> (x,y+40)

so

The new function will be

[tex]f(x)=5x+10+40[/tex]

[tex]f(x)=5x+50[/tex]

The translation of the original linear function, f(x) = 5·x + 50, gives the

following values;

Part A:

Part B:

Part C:

The function for the amount of money in the account is; f(x) = 5·x + 50

Part A

1. The graph of the translation of the above function up 10 units gives

the function;

f(x) + 10 = 5·x + 50 + 10 = 5·x + 60

f(x) + 10 = 5·x + 60

Please find attached the graph of the function translated up 10 units created with MS Excel

2. The coordinate rule for a translation up 10 units is; [tex]\underline{(x, \ y + 10)}[/tex]

3. The meaning of the translation up 10 units means that amount in the

account at a point in time is increased by $10

Part B;

1. The function, f(x) = 5·x + 50, translated 10 units to the right gives;

f(x + 10) = 5·(x + 10) + 50 = 5·x + 100

Please find attached the graph of the function translated right 10 units created with MS Excel

2. The coordinate rule is (x, y) [tex]\underrightarrow{T_{(10, \ 0)}}[/tex] [tex]\underline{(x + 10, \ y)}[/tex]

3. A translation of the function to the right, means that the point in time at

which the graph starts, the account balance is $100, such that Jeremy

the time the interest is applied is 10 years longer, than the original time

added to the number of years in the given function, f(x) = 5·x + 50

Part C

1. The characteristic of the graph that stays the same is the slope

2. The vertical translation in the graph of the translation right 10 units

compared to the original graph is 50 units.

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Answer:

Don't know if this is correct but, I think the answer is 3/4.

Answer:

[tex]\dfrac{3}{4}[/tex]

Step-by-step explanation:

The given expression is

[tex]a^3b^{-2}c^{-1}d[/tex]

We need to find the value of this expression in reduced fraction if a=2 b=4 c=10 d=15.

Substitute a=2 b=4 c=10 d=15 in given expression.

[tex](2)^3(4)^{-2}(10)^{-1}(15)[/tex]

Using the property of exponent, we get

[tex](2)^3\times \left(\dfrac{1}{4^2}\right)\times \left(\dfrac{1}{10}\right)\times (15)[/tex]         [tex][\because a^{-n}=\dfrac{1}{a^n}][/tex]

[tex]8\times \left(\dfrac{1}{16}\right)\times \left(\dfrac{1}{10}\right)\times (15)[/tex]

Cancel out common factors.

[tex]1\times \left(\dfrac{1}{2}\right)\times \left(\dfrac{1}{2}\right)\times (3)[/tex]

[tex]\dfrac{3}{4}[/tex]

Hence, the required fraction is 3/4.

Answer:

Angle measurements of 35°, 35°, and 110°

Angle measurements of 40°, 60°, and 80°

Step-by-step explanation:

If you specify only the three angles, you can have an infinite number of similar triangles of different sizes.

A is wrong. There can be only one triangle in which all three side lengths are specified.

C is wrong. The angles do not add up to 180°.

The descriptions which can describe more than one triangle is:

a)

Side lengths 6 ft, 8 ft and 10 ft.

First we check whether it form a triangle or not:

6+8=14>10

8+10=18>6

and

6+10=16>8

Yes, these side lengths form a triangle.

With the help of this only one unique triangle is possible.

b)

angle measurements of 35°,35° and 110°

First we check the sum of the angles:

35°+35°+110°=180°

Hence, more than one triangle is possible.

c)

angle measurements of 30°,40° and 50°

First we check the sum of these angles:

30°+40°+50°=120°≠180°

Hence, these angles measure do not form a triangle.

d)

angle measurements of 40°,60° and 80°

First we check the sum of these angles:

40°+60°+80°=180°.

Hence, more than one triangle is possible.

e)

side length of 4 cm, 6 cm and 9 cm.

First we check whether it form a triangle or not:

4+6=10>9

6+9=15>4

and

4+9=13>6

Hence, a unique triangle will be formed.

Answer:

[tex]3i \sqrt{7}[/tex]

Step-by-step explanation:

The imaginary unit is [tex]i=\sqrt{-1}[/tex].

Now 63 itself is not a perfect square but 63 does contain a factor that is.

63=9×7 and 9 is a perfect square.

So [tex]\sqrt{-63}=i \sqrt{63}[/tex].

[tex]i \sqrt{63}=i \sqrt{9 \cdot 7}=i \sqrt{9} \sqrt{7}=i(3) \sqrt{7}=3i \sqrt{7}[/tex]

To find the square root of -63, you'll need to consider the concept of imaginary numbers because the square root of a negative number is not a real number. Here's how you can find the equivalent expression:
The square root of -63 can be expressed as √(-63). Since you cannot take the square root of a negative number in the real number system, you would use an imaginary unit, which we designate as "i". The imaginary unit "i" is defined as √(-1).
Now, you can factor -63 into -1 and 63, so the expression becomes:
√(-63) = √(-1 * 63)
This simplifies to:
√(-1) * √(63)
We already established that √(-1) is represented by the imaginary unit "i". So now the expression is:
i * √(63)
The √(63) doesn't simplify neatly into a whole number since 63 is not a perfect square. However, 63 can be factored into 9 and 7, where 9 is a perfect square. Let's do that:
√(63) = √(9 * 7)
√(63) = √(9) * √(7)
√(63) = 3 * √(7)
Therefore, the expression now looks like this:
i * 3 * √(7)
Since multiplication is commutative, you can reorder this:
3 * i * √(7)
Thus, the expression 3 * i * √(7) is equivalent to √(-63). This is because 3 * √(7) is the real number part and "i" indicates that it is an imaginary number due to the original square root of a negative number.

Answer:

x = 1/3 and y = 3

Step-by-step explanation:

It is given the system of equations

3x + 2y = 7   ----(1)

-3x + 3y = 8    ----(2)

To find the solution

Add eq(1) and eq (2) we get,

3x + 2y = 7   ----(1)

-3x + 3y = 8    ----(2)

 0 + 5y = 15

y = 15/5 = 3

Substitute the value of y in eq(1)

3x + 2y = 7   ----(1)

3x + 2*3 = 7

3x + 6 = 7

3x = 7 - 6 = 1

x = 1/3

Therefore

x = 1/3 and y = 3

Answer:

The domain would be 0 to 4 seconds

Step-by-step explanation:

According to the given conditions a ball was lying on the ground but when it was hit by Vanessa it was in the air for 4 seconds. We have to find the domain.

We know that the domain is the input values. According to the given statement it would start at time 0 and end when it hit the ground at 4 seconds.

Therefore the domain would be 0 to 4 seconds.

[0,4] ....

Answer:

I think 12.

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

Using the substitution u = [tex]x^{\frac{1}{2} }[/tex]

noting that ([tex]\frac{1}{2}[/tex] )² = [tex]\frac{1}{4}[/tex], then

[tex]x^{\frac{1}{2} }[/tex] + 9[tex]x^{\frac{1}{4} }[/tex] + 20 = 0

Can be rewritten as

u + 9u² + 20 = 0, that is

9u² + u + 20 = 0 ← in standard form

Answer:

C. 16

Step-by-step explanation:

The distance between two points on the number line is the absolute value of the difference of their coordinates.

If the coordinates of two points on the number line are a and b, then the distance between the two points is

d = |a - b|

The coordinates here are -9 and 7.

distance = |-9 - 7| = |-16| = 16

Answer:

A

Step-by-step explanation:

we have g(x) = 5x+3  

we know that 5x=f(x)

so g(x)=f(x)+3  

thus g(x) is obtained by adding 3 units to f(x)  

that's means the graph of g is obtained by translating the parent function f by 3 units up

Answer:

Step-by-step explanation:

There are two functions f(x) = 5x and g(x) = 5x + 3

This parent function f(x) has been translated to form a new function g(x) = 5x + 3

These linear functions can be represented by two lines y = 5x and y' = 5x' + 3

So when we shift line y = 5x by 3 units up on the y-axis then a new graph of

y = 5x + 3 will be formed.

Therefore, Option A is the answer.

Answer:

x = 50

Step-by-step explanation:

The sum of the 3 angles in a triangle = 180°

Sum the 2 given angles in the lower triangle and subtract from 180 to find the third angle

third angle = 180° - (65 + 65)° = 180° - 130° = 50°

x = third angle = 50 ( vertical angles )

A triangle has to have three angles and they have to add up to 180 degrees so,

65+65=130

180-130=50

X=50 degrees because it is equal to the third angle.

6x + 2 + 40 = 90

Let x = the unknown

6x + 42 = 90

6x = 90 - 42

6x = 48

x = 48/6

x = 8

Answer:

x=8

Step-by-step explanation:

We know that 40 + (6x+2)  +90 = 180   since the three angles form a straight line

40 + (6x+2)  +90 = 180

Combine like terms

132 + 6x = 180

Subtract 132 from each side

132-132 +6x= 180-132

6x = 48

Divide each side by 6

6x/6 = 48/6

x = 8

Answer:

9/64

Step-by-step explanation:

The probability that the marble on the first draw is blue is 3/8.

The probability that the marble on the second draw is blue is also 3/8.

So the probability that both are blue is:

3/8 × 3/8 = 9/64 ≈ 14%

The probability that both of the marbles drawn from the bag are blue is 9/64.

Probability is the chance that an event would occur. The probability the event occurs is 1 and the probability that the event does not occur is 0.

The probability that both of the marbles drawn are blue = fraction of blue marbles²

= 3/8 x 3/8 = 9/64

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Answer:

$57.75

Step-by-step explanation:

If 11 books cost $42.35,

11b=42.35

b=42.35/11

b=3.85

15 books will cost 15b.

15b

= 15*3.85

= 57.75.

15 Books will cost $57.75.

Answer:

57.75

Step-by-step explanation:

First I found the unit rate by dividing 42.35 by 11, which is 3.85.

second i multiplied the unit rate by 15 and got 57.75

a system of linear equations with infinite solutions, is simply one that has the same equation twice, but but but, one of the equations is in disguise.

so, say we can just  hmmm multiply the coefficient of the "x" variable, which is the slope, by something that gives us 1, recall same/same = 1, hmmm say let's multiply it by hmmmm 7/7.

[tex]\bf y=\cfrac{2}{3}x-5\implies \stackrel{\textit{multiplying the slope by }\frac{7}{7}}{\cfrac{2}{3}\cdot \cfrac{7}{7}\implies \cfrac{14}{21}}\implies \stackrel{\textit{so we get this equation}}{y=\cfrac{14}{21}x-5}[/tex]

now, let's notice that 14/21 simplifies to 2/3, so is really the same slope and the same y-intercept.

so if we use those two equations in a system of equations and graph them, what happens is, the first one will graph a line, the second one will graph another line BUT right on top of the first one drawn, so the two lines will just be pancaked on top of each other, making every point in each line, "a solution", since they're meeting at every point, and since lines go to infinite, "infinitely many solutions".

Answer:

Sample Response/Explanation: To have an infinite number of solutions, the equations must graph the same line. That means the equations must be equivalent. To form an equivalent equation, use the properties of equality to rewrite the given equation in a different form. Add, subtract, multiply, or divide both sides of the equation by the same amount.

Step-by-step explanation:

Answer:

So length is 7 in while width is 1 in.

Step-by-step explanation:

We are given W is 6 inches less than L which mean as an equation we have W=L-6.

We are given the area of this rectangle, LW=7.

So we have the system:

W=L-6

LW=7.

Replace the second W with what the first W equals:

LW=7

L(L-6)=7

Distribute:

[tex]L^2-6L=7[/tex]

Subtract 7 on both sides:

[tex]L^2-6L-7=0[/tex]

We are luck since the coefficient of L^2 is 1.  This means all we have to do is find two numbers that multiply to be -7 add at the same time add up to -6.

Those numbers are -7 and 1 since (-7)(1)=-7 and (-7)+(1)=-6.

So the factored form of our equation is:

(L-7)(L+1)=0

This gives us two equations to solve:

L-7=0  or L+1=0

L=7     or L=-1

L=-1 doesn't make sense for a length so L=7.

L=7 means the length is 7 inches.

If W=L-6 and L=7, then W=7-6=1.

The width is 1 inch since W=1.

So length is 7 in while width is 1 in.

Answer:

The ordered pair (3,1/2) is not a common point, the statement is false

The solution is the point (0.667, 6.333) (common point)

Step-by-step explanation:

we have

[tex]y=2x+5[/tex] -----> equation A

[tex]y=\frac{1}{2}x+6[/tex] ----> equation B

we know that

If a ordered pair is a common point between the two lines,

then

the ordered pair must satisfy both equations

step 1

Verify if the ordered pair (3,1/2) satisfy equation A

substitute the value of x and the value of y in the equation A and then compare the results

For x=3, y=1/2

[tex]\frac{1}{2}=2(3)+5[/tex]

[tex]\frac{1}{2}=11[/tex] ----> is not true

The ordered pair don't satisfy the equation A

therefore

The ordered pair is not a common points both lines

The statement is false

step 2

Find the common point between the two lines

[tex]y=2x+5[/tex] -----> equation A

[tex]y=\frac{1}{2}x+6[/tex] ----> equation B

Solve the system of equations by graphing

The intersection point both graphs is the solution of the system (common point)

using a graphing tool

The solution is the point (0.667, 6.333) (common point)

see the attached figure

Answer:

1.19x=12

Step-by-step explanation:

The 12 represents his budget and 1.19 is the cost of each song, x is the amount of songs.

Answer: [tex]1.19x=12[/tex]

Step-by-step explanation:

Given : Doug can download new songs for $1.19 each.

Let the number of songs downloaded be x .

Then the total cost of x songs will be :-

[tex]1.19x[/tex]

To find the number of songs he can download for $12.00 , we need to put [tex]1.19x[/tex] equals to 12.

We get, the equation to show how many songs he can download for $12.00 will be

[tex]1.19x=12[/tex]

Answer:

so b=-9 while c=12.

Step-by-step explanation:

If you have x-intercepts 1 and -4, then that means f(1)=0 and f(-4)=0.

You are given [tex]f(x)=-3x^2+bx+c[/tex]

So you have the system of equations to solve:

[tex]f(1)=-3(1)^2+b(1)+c=0[/tex]

[tex]f(-4)=-3(-4)^2+b(-4)+c=0[/tex]

Evaluating the exponents:

[tex]-3(1)+b+c=0[/tex]

[tex]-3(16)-4b+c=0[/tex]

Doing a little bit of multiplying:

[tex]-3+b+c=0[/tex]

[tex]-48-4b+c=0[/tex]

Let's add 3 on both sides of equation 1 and 48 on both sides of equation 2:

[tex]b+c=3[/tex]

[tex]-4b+c=48[/tex]

Subtracting the equations will eliminate c.

Let's do that:

[tex]5b+0c=-45[/tex]

[tex]5b=-45[/tex]

Divide both sides by 5:

[tex]b=\frac{-45}{5}[/tex]

Simplify:

[tex]b=-9[/tex]

If b=-9 and b+c=3 then -9+c=3 implies c=9+3=12.

so b=-9 while c=12.

To find b and c for the parabola y=-3x^2+bx+c with x-intercepts 1 and -4, we express the equation in its factored form, then expand it. From the expansion, we see that b=-9 and c=-12.

To find the values of b and c in the equation y = -3x^2+bx+c, where the x-intercepts are given as 1 and -4, we use the following steps:

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Answer:

A. 36

Step-by-step explanation:

65+79=144

180-144=36

Answer: A

Step-by-step explanation: