The box plots show the weights, in pounds, of the dogs in two different animal shelters.

Weights of Dogs in Shelter A (Top Plot)

Weights of Dogs in Shelter B(Bottom Plot)

Which correctly compares the ranges of the data?
The range in shelter A is 11, and the range in shelter B is 4.
The range in shelter A is 20, and the range in shelter B is 10.
The range in shelter A is 13, and the range in shelter B is 8.
The range in shelter A is 22, and the range in shelter B is 18.

Answer:

Time taken by Doreen is 6 hours and speed is 10 miles per hour.

Time taken by Sue is 4 hours and speed is 20 miles per hour.

Step-by-step explanation:

Let the speed of Doreen be x

According to the question  speed of Sue   is = x+10

time  taken By Sue to cover 80 miles = [tex]\frac{80}{x+10}[/tex]

time taken by Doreen to travel 60 miles = [tex]\frac{60}{x}[/tex]

According to question Sue take two hours less than Doreen takes

therefore

[tex]\frac{60}{x}[/tex] - [tex]\frac{80}{x+10}[/tex] =2

[tex]\frac{60(x+10)-80x}{x(x+10)}[/tex] =2

60(x+10) -80x = 2(x(x+10)

60x+600-80x = [tex]2x^2+20x\\[/tex]

simplifying it ,we get

[tex]2x^2+40x-600=0\\[/tex]

Dividing both sides by 2 ,we get

[tex]x^2+20x-300=0\\[/tex]

solving it for x ,we get

(x+30)(x-10) =0

x =-30 which is not possible

x =10 miles per hour

Speed of Doreen = 10 miles per hour

Speed of Sue = 10+10 = 20 miles per hour

Time taken by Doreen = 60 divided by 10 = 6 hours

Time taken  by Sue = 80 divided by 20 = 4 hours

In conclusion, Doreen travels at a speed of 30 mph, taking her 4 hours to travel 60 miles. Sue, on the other hand, travels at a speed of 40 mph, taking her 2 hours to travel 80 miles.

This problem is a classic example of distance, rate, and time relations in mathematics. Let's start by denoting Sue's speed as x mph, the Doreen's speed would then be x-10 mph. We know that time is equal to distance divided by speed. So, the time it takes Sue to travel 80 miles would be 80/x hours and the time it takes Doreen to travel 60 miles would be 60/(x-10) hours. The question states that Sue's travel time is 2 hours less than Doreen's. Therefore, we can form the equation: 60/(x-10) = 80/x + 2. Solving this equation, we find that x equals 40 mph, which is Sue's speed and Doreen's speed is 30 mph. Consequently, the time it takes Sue to travel 80 miles is 2 hours and for Doreen to travel 60 miles is 4 hours.

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Answer:

a. 0.17

Step-by-step explanation:

Total number of light bulbs = 8

The probability that each individual light bulb works = 0.8

The working of light bulbs is independent of each other, this means one light bulb does not influence the other light bulbs.

We need to calculate the probability that all eight of the light bulbs work. Since the light bulbs work independently, the overall probability of independent events occurring together is the product of their individual probabilities. Therefore,

Probability that all eight of the light bulbs work = 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8

= [tex](0.8)^{8}[/tex]

= 0.16777216

≈ 0.17

Thus, option a gives the correct probability that all eight of the light bulbs work

You can use binomial distribution, and thus, its probability function to find the needed probability.

The probability that all eight of the light bulbs work is 0.167

Binomial distributions consists of n independent Bernoulli trials.

Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))

Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as

[tex]X \sim B(n,p)[/tex]

The probability that out of n trials, there'd be x successes is given by

[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]

Since all the light bulbs' working is independent, and each bulb's chance of working is 0.8 and there are 8 bulbs, thus,

n = 8

p = 0.8

and Let X be a random variable tracking how many out of 8 bulbs are working, then we have:

[tex]X \sim B(8, 0.8)[/tex]

Then, the needed probability is P(X = 8) (since we need to know probability that all 8 bulbs will work)

By using the probability mass function of binomial distribution, we get:

[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}\\P(X = 8) = \:^8C_8(0.8)^8(1-0.8)^{8-8} = 1 \times (0.8)^8 \times 1 \approx 0.167[/tex]

Thus,

The probability that all eight of the light bulbs work is 0.167

Learn more about binomial distribution here:

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Answer:

(3,4,5)

(6,8,10)

(5,12,13)

(8,15,17)

(12,16,20)

(7,24,25)

(10,24,26)

(20,21,29)

(16,30,34)

(9,40,41)

Just choose 2 numbers from {1,2,3,4,5,6,7,8,...} and make sure the one you input for x is larger.

Post the three in the comments and I will check them for you.

Step-by-step explanation:

We need to choose 2 positive integers for x and y where x>y.

Positive integers are {1,2,3,4,5,6,7,.....}.

I'm going to start with (x,y)=(2,1).

x=2 and y=1.

[tex](2^2+1^2)^2=(2^2-1^2)^2+(2\cdot2\cdot1)^2[/tex]

[tex](4+1)^2=(4-1)^2+(4)^2[/tex]

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

So one Pythagorean Triple is (3,4,5).

I'm going to choose (x,y)=(3,1).

x=3 and y=1.

[tex](3^2+1^2)^2=(3^2-1^2)^2+(2\cdot3\cdot1)^2[/tex]

[tex](9+1)^2=(9-1)^2+(6)^2[/tex]

[tex](10)^2=(8)^2+(6)^2[/tex]

So another Pythagorean Triple is (6,8,10).

I'm going to choose (x,y)=(3,2).

x=3 and y=2.

[tex](3^2+2^2)^2=(3^2-2^2)^2+(2\cdot3\cdot2)^2[/tex]

[tex](9+4)^2=(9-4)^2+(12)^2[/tex]

[tex](13)^2=(5)^2+(12)^2[/tex]

So another is (5,12,13).

I'm going to choose (x,y)=(4,1).

[tex](4^2+1^2)^2=(4^2-1^2)^2+(2\cdot4\cdot1)^2[/tex]

[tex](16+1)^2=(16-1)^2+(8)^2[/tex]

[tex](17)^2=(15)^2+(8)^2[/tex]

Another is (8,15,17).

I'm going to choose (x,y)=(4,2).

[tex](4^2+2^2)^2=(4^2-2^2)^2+(2\cdot4\cdot2)^2[/tex]

[tex](16+4)^2=(16-4)^2+(16)^2[/tex]

[tex](20)^2=(12)^2+(16)^2[/tex]

We have another which is (12,16,20).

I'm going to choose (x,y)=(4,3).

[tex](4^2+3^2)^2=(4^2-3^2)^2+(2\cdot4\cdot3)^2[/tex]

[tex](16+9)^2=(16-9)^2+(24)^2[/tex]

[tex](25)^2=(7)^2+(24)^2[/tex]

We have another is (7,24,25).

You are just choosing numbers from the positive integer set {1,2,3,4,... } and making sure the number you plug in for x is higher than the number for y.

I will do one more.

Let's choose (x,y)=(5,1).

[tex](5^2+1^2)^2=(5^2-1^2)^2+(2\cdot5\cdot1)^2[/tex]

[tex](25+1)^2=(25-1)^2+(10)^2[/tex]

[tex](26)^2=(24)^2+(10)^2[/tex]

So (10,24,26) is another.

Let (x,y)=(5,2).

[tex](5^2+2^2)^2=(5^2-2^2)^2+(2\cdot5\cdot2)^2[/tex]

[tex](25+4)^2=(25-4)^2+(20)^2[/tex]

[tex](29)^2=(21)^2+(20)^2[/tex]

So another Pythagorean Triple is (20,21,29).

Choose (x,y)=(5,3).

[tex](5^2+3^2)^2=(5^2-3^2)^2+(2\cdot5\cdot3)^2[/tex]

[tex](25+9)^2=(25-9)^2+(30)^2[/tex]

[tex](34)^2=(16)^2+(30)^2[/tex]

Another Pythagorean Triple is (16,30,34).

Let (x,y)=(5,4)

[tex](5^2+4^2)^2=(5^2-4^2)^2+(2\cdot5\cdot4)^2[/tex]

[tex](25+16)^2=(25-16)^2+(40)^2[/tex]

[tex](41)^2=(9)^2+(40)^2[/tex]

Another is (9,40,41).

Answer:

[tex]AC=4.3\ in[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The triangle AOC is an isosceles triangle

OA=OC=5/2=2.5 in -----> the radius of the circle

∠AOC=180°-60°=120°

∠CAO=∠ACO=120°/2=60°

Applying the law of cosines find the length of the chord AC

[tex]AC^{2}=OA^{2}+OC^{2}-2(OA)(OC)cos(120\°)[/tex]

substitute

[tex]AC^{2}=2.5^{2}+2.5^{2}-2(2.5)(2.5)cos(120\°)[/tex]

[tex]AC^{2}=18.75[/tex]

[tex]AC=4.3\ in[/tex]

Answer:

E) -3, 5

Step-by-step explanation:

x – 15 / x = 2

x^2 - 15 = 2x

x^2 - 2x - 15 = 0

(x - 5)(x + 3) = 0

x - 5 = 0; x = 5

x + 3 = 0; x = -3

Solutions: -3, 5

For this case we must solve the following equation:

[tex]x- \frac {15} {x} = 2[/tex]

We manipulate the equation algebraically:

[tex]\frac {x ^ 2-15} {x} = 2\\x ^ 2-15 = 2x\\x ^ 2-2x-15 = 0[/tex]

To solve, we factor the equation. We must find two numbers that when multiplied by -15 and when summed by -2. These numbers are:

+3 and -5.

[tex](x + 3) (x-5) = 0[/tex]

So, the roots are:

[tex]x_ {1} = - 3\\x_ {2} = 5[/tex]

Answer:

Option E

Answer:

  B.  46,368

Step-by-step explanation:

Each Fibonacci number is the sum of the previous two. The next one is the sum of the two that are given.

  F(24) = F(22) +F(23) = 17,711 +28,657 = 46,368

Answer:

0.889

Step-by-step explanation:

I think it is multiplied.

10 *9 = 90

which is the total number of ways you can draw 2 cards without replacement.

I think it is easier to figure out how many possibilities there are over 62 and go from there.

10*9

10*8

10*7

===========

9*8

9*7

So there are 5 combinations that are over 62. There are 5 more possibilities because you could draw them in the reverse order

9 * 10

8 * 10

7 * 10  

8* 9

7 * 9

In all there are 10 ways of drawing numbers that are over 62

So what is the probability of drawing 2 cards above 62?

10/90 = 1/9 = 0.111

Therefore, there must be a probability of 1 - 0.111 for under 62 = 0.889

Answer:

2 gallons

Step-by-step explanation:

At this speed, the car uses 1 gallon of fuel for a distance of 25 miles.

We need the number of miles the car travels in 75 minutes to find the amount of fuel it uses.

75 minutes * (1 hour)/(60 minutes) = 1.25 hours

speed = distance/time

distance = speed * time

distance = 40 miles/hour * 1.25 hours = 50 miles

In 75 minutes, at 40 mph, the car travels 50 miles.

(1 gal)/(25 miles) = x/(50 miles)

x = 2 gal

Answer: 2 gallons

The car will use 2 gallons of fuel when traveling at a constant speed of 40 miles per hour for 75 minutes.

To find the number of gallons of fuel used when the car travels at a constant speed of 40 miles per hour for 75 minutes, we can use the formula:

Gallons of fuel used = (Distance traveled in miles) / (Miles per gallon)

Since the car travels at a constant speed of 40 miles per hour, it covers a distance of 40 miles in 1 hour. Therefore, in 75 minutes it will travel 40 miles * (75 minutes / 60 minutes per hour) = 50 miles.

Now, we can calculate the number of gallons of fuel used: Gallons of fuel used = 50 miles / 25 miles per gallon = 2 gallons.

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Answer:

Option D

Step-by-step explanation:

We have the following variable definitions:

sofas: x

pillows: y

Pillows come in pairs so we have 2y pillows

The total order for all the possible combinations is:

[tex]x+2y[/tex]

The wholesaler requires a minimum of 4 items in each order from its retail customers. This means the retailers can order 4 or more.

Therefore the inequality is:

[tex]x+2y\ge4[/tex]

To graph this inequality, we graph the corresponding linear equation, [tex]x+2y=4[/tex]  with a solid line and shade above.

The correct choice is D

See attachment

Answer:

 55

Step-by-step explanation:

Let x represent the middle integer. Then the smallest is x-2 and the largest is x+2. Your requirement is that ...

  (x-2)/(x+2) > 2/3

  3x -6 > 2x +4 . . . . cross multiply

  x > 10 . . . . . . . . . . .add 6-2x

The smallest integer satisfying this requirement is x=11. The sum of the 5 integers is 5x = 55.

The smallest sum is 55.

Answer:

55

Step-by-step explanation:

Answer:

  x = 117°, 297°

Step-by-step explanation:

Subtract 1 from both sides of the equation and you have ...

  tan(x) = -2

Then the arctangent function tells you ...

  x = arctan(-2) ≈ 116.5651°, 296.5651°

  x ≈ 117° or 297°

The maximum difference in the lengths of the radii is 1/2.

To solve this problem, let's denote the radius of the smaller circle as ( r ) and the radius of the larger circle as ( R ). We're given that [tex]\( r + R = 10 \)[/tex].

The area of a circle is given by the formula [tex]\( A = \pi r^2 \)[/tex], where ( r ) is the radius.

We want the absolute difference in the areas of the two circles to be less than or equal to [tex]\( 5\pi \)[/tex]. So, we can set up the following inequality:

[tex]\[ |(\pi R^2) - (\pi r^2)| \leq 5\pi \][/tex]

[tex]\[ |(\pi (10-r)^2) - (\pi r^2)| \leq 5\pi \][/tex]

Expanding and simplifying:

[tex]\[ |(100\pi - 20\pi r + \pi r^2) - (\pi r^2)| \leq 5\pi \][/tex]

[tex]\[ |100\pi - 20\pi r| \leq 5\pi \][/tex]

[tex]\[ 100 - 20r \leq 5 \][/tex]

[tex]\[ 100 - 5 \leq 20r \][/tex]

[tex]\[ 95 \leq 20r \][/tex]

[tex]\[ \frac{95}{20} \leq r \][/tex]

[tex]\[ r \geq \frac{19}{4} \][/tex]

So, the maximum difference in the lengths of the radii is when [tex]\( r = \frac{19}{4} \)[/tex] and [tex]\( R = 10 - r = 10 - \frac{19}{4} = \frac{21}{4} \)[/tex].

The maximum difference in the lengths of the radii is [tex]\( \frac{21}{4} - \frac{19}{4} = \frac{2}{4} = \frac{1}{2} \)[/tex].

Answer:

Given:

Diameter of lake = 2 miles

∴ [tex]Radius = \frac{Diameter}{2}[/tex] = 1 miles  

The circumference of the lake can be computed as :

Circumference = 2πr

Circumference = 2×3.14×1 = 6.28 miles

This circumference is the total distance traveled by Johanna.

We are give the speed at which Johanna jogs, i.e. Speed = 3 miles/hour

∴ Time taken by Johanna to jog around the lake is given as :

[tex]Time = \frac{Distance}{Speed}[/tex]

Time = 2.093 hours

∴ The correct option is (c.)

Answer:

Step-by-step explanation:

Part A

xf = xo + vo* t + 1/2 a*t^2                    Subtract xo

xf - xo = 0*t + 1/2 a*t^2                        multiply by 2

2(xf - xo) = at^2                                    divide by t^2

2(xf - xo ) / t^2 = a

Part B

Givens

xo =0

vo = 0

a = 10  m/s^2

xf = 120 m

Solution

xf = xo + vo* t + 1/2 a*t^2              Substitute the givens

120 = 0 + 0 + 1/2 * 10 * t^2            Multiply by 2

120*2 = 10* t^2                  

240 = 10*t^2                                  Divide by 10

240/10  = t^2

24 = t^2                                          take the square root of both sides.

√24 = √t^2

t = √24

t = √(2 * 2 * 2 * 3)

t = 2√6

Answer:

=20s³+50s²+32s+6

Step-by-step explanation:

We multiply each of the term in the initial expression by the the second expression as follows:

4s(5s²+10s+3)+2(5s²+10s+3)

=20s³+40s²+12s+10s²+20s+6

Collect like terms together.

=20s³+50s²+32s+6

If its acceleration is constant, then it is equal to the jet's average velocity, given by

[tex]a=a_{\rm ave}=\dfrac{\Delta v}{\Delta t}[/tex]

Then it takes

[tex]17.7\dfrac{\rm m}{\mathrm s^2}=\dfrac{233\frac{\rm m}{\rm s}-119\frac{\rm m}{\rm s}}{\Delta t}\implies\Delta t=\boxed{6.44\,\mathrm s}[/tex]

Answer:

The time taken by the jet is 6.44 seconds.

Step-by-step explanation:

It is given that,

Acceleration of the jet, [tex]a=17.7\ m/s^2[/tex]

Initial velocity of the jet, u = 119 m/s

Final velocity of the jet, v = 233 m/s

Acceleration of an object is given by :

[tex]a=\dfrac{v-u}{t}[/tex]

[tex]t=\dfrac{v-u}{a}[/tex]

[tex]t=\dfrac{233-119}{17.7}[/tex]

t = 6.44 seconds

So, the time taken by the jet is 6.44 seconds. Hence, this is the required solution.

Answer:

Component form of u is (-18,13)

The magnitude of u is 22.2

Step-by-step explanation:

The component form of a vector is an ordered pair that describe the change is x and y values

This is mathematically expressed as (Δx,Δy) where Δx=x₂-x₁ and Δy=y₂-y₁

Given ;

Initial points of the vector as (14,-6)

Terminal point of the vector as (-4,7)

Here x₁=14,x₂=-4, y₁=-6 ,y₂=7

The component form of the vector u is (-4-14,7--6) =(-18,13)

Finding Magnitude of the vector

║u=√(x₂-x₁)²+(y₂-y₁)²

║u=√-18²+13²

║u=√324+169

║u=√493

║u=22.2

Answer:

Answer would be 9y+36

Step-by-step explanation:

Because if you distribute the 9 inside the parenthesis, you'd get

9*y=9y and 9*4=36

so 9y+36

Hope my answer was helpful to you!

Final answer:

The Distributive Property is used to rewrite the expression 9(y + 4) as 9y + 36 by multiplying 9 by each term inside the parentheses.

Explanation:

To use the Distributive Property to rewrite the expression 9(y + 4), we would distribute the number 9 to both y and 4 inside the parentheses. This means we multiply 9 by y and then multiply 9 by 4, combining the results with the addition operations between them.

Using the distributive property, we get:

9 times y = 9y

9 times 4 = 36

So, the expression will be rewritten as:

9y + 36

Therefore, by distributing the 9, we have turned the original expression into a sum of two terms, which are a number, variable, or a product/quotient of numbers and/or variables separated by + or - signs. In this case, the terms are 9y and 36.

Answer:

[tex]\frac{\pi }{6}[/tex]

Step-by-step explanation:

The volume of a sphere is [tex]\frac{4}{3} \pi r^{3}[/tex]

Just plug in 1/2 for r

[tex]\frac{4}{3} \pi (\frac{1}{2}) ^{3}[/tex]

The answer is [tex]\frac{\pi }{6}[/tex]

Answer:

  C'(4, 4)

Step-by-step explanation:

We assume dilation is about the origin, so all coordinates are multiplied by the scale factor:

  C' = 2C = 2(2, 2) = (4, 4)

Answer:

  (y² +1)/y

Step-by-step explanation:

Invert the denominator fraction and multiply. Factor the difference of squares.

[tex]\displaystyle\frac{\left(\frac{6y^2-6}{8y^2+8y}\right)}{\left(\frac{3y-3}{4y^2+4}\right)}=\frac{6(y^2-1)}{8y(y+1)}\cdot\frac{4(y^2+1)}{3(y-1)}\\\\=\frac{24(y+1)(y-1)(y^2+1)}{24y(y+1)(y-1)}=\frac{y^2+1}{y}[/tex]

Answer:

b. y+7=5x

Step-by-step explanation:

a. 1-3x^2     is a quadratic

b. y+7=5x    is a linear function:  y = 5x - 7

c. x^3 + 4 = y   is a cubic function

d. 9(x^2-y) = 3    is a quadratic function

e.y-x^3=8   is a cubic function

Answer:

Step-by-step explanation:

f(10) = 120 tells you that after x = 10 minutes, the coffee is 120 degrees

The statement f(10)=120 indicates that after 10 minutes, the coffee's temperature is 120 degrees Fahrenheit.

When we see an equation such as f(10)=120, it tells us that after 10 minutes, the temperature of the coffee has cooled down to 120 degrees Fahrenheit. The function f(x) describes the temperature of the coffee after x minutes, so the specific point f(10)=120 provides us with a snapshot of the temperature at that particular time.

4(x+2)=3(x-2)

Multiply the first bracket by 4

Multiply the second bracket by 3

4x+8=3x-6

Move 3x to the left hand side, whenever moving a number with a letter the sign changes ( positive 3x to negative 3x)

4x-3x+8=3x-3x-6

x+8=-6

Move positive 8 to the right hand side

x+8-8=-6-8

x=-14

Check answer by using substitution method

Use x=-14 into both of the equations

4(-14+2)=3(-14-2)

-56+8=-42-6

-48=-48

Answer is -14- D)

The algebraic equation 4(x + 2) = 3(x − 2) is solved by distribution, combining like terms, and isolating the variable x, which results in x = -14.

This is a simple algebraic equation problem. We solve 4(x + 2) = 3(x − 2) by following these steps:

So, x = -14 is the solution.

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Circle theorem:

The angle at the centre (T) is double the angle at the circumference (Q)

---> That also means that:

The angle at the circumference (Q) is half the angle at the centre (T)

Since T = 130 degrees;

Q = 130 divided by 2

   = 65°

___________________________________

Answer:

∠Q = 65°

Answer:

m<Q = 65°

Step-by-step explanation:

It is given that <T = 130°

To find the <Q

From the figure we can see that <T is the central angle made by the arc RS

And <Q is the angle made by the arc RS on minor arc.

We know that m<Q = (1/2)m<T

We have m<T = 130°

Therefore m<Q = 130/2 = 65°

Answer:

[tex]50xy[/tex] and [tex]40x^2[/tex].

Step-by-step explanation:

The given polynomial is [tex]36x^3-22x^2-144x[/tex].

The prime factorization of each term are;

[tex]36x^2=2^2\times 3^2\times x^3[/tex]

[tex]-22x^2=-2\times 11\times x^2[/tex]

[tex]-144x=-2^4\times 3^2\times x[/tex]

The greatest common factor of these three terms is [tex]2x[/tex].

Now observe that:

The GCF of [tex]2x[/tex] and 11 is 1

The GCF of [tex]2x[/tex] and 50xy is 2x

The GCF of [tex]2x[/tex] and [tex]40x^2[/tex] is 2x

The GCF of [tex]2x[/tex] and 24 is 2

The GCF of [tex]2x[/tex] and 10y is 2

The correct options are [tex]50xy[/tex] and [tex]40x^2[/tex].

Answer:

Option 4: (x+1)^2+(y-1)^2 = 16

Step-by-step explanation:

The radius of the given circle in attached picture is: 4 units

The center is denoted by (h,k) = (-1,1)

So,

The standard form of equation with center at (h,k) and radius r

(x-h)^2 + (y-k)^2 = r^2

Putting the values

(x-(-1))^2 + (y-1)^2 = 4^2

(x+1)^2+(y-1)^2 = 16

Hence option number 4 is correct ..

Answer:

Length of arc AB is,

= 2πr (angle between AB) /360

=2×3.14×90/360

=1.57 cm

For this case we have that by definition, the arc length of a circle is given by:

[tex]AL = \frac {x * 2 \pi * r} {360}[/tex]

Where:

x: Represents the angle between AB. According to the figure we have that x = 90 degrees.

[tex]r = 7.9 \ cm[/tex]

So:

[tex]AL = \frac {90 * 2 \pi * 7.9} {360}\\AL = \frac {90 * 2 * 3.14 * 7.9} {360}\\AL = \frac {4465,08} {360}\\AL = 12.403[/tex]

Answer:

[tex]12.4\ cm[/tex]

Answer:

2x^3+8x^2-54x-84

Step-by-step explanation:

Answer:

2(x + 7)(x² - 3x - 6) = 2x³ + 8x² - 54x - 84

Step-by-step explanation:

Simplification is a method used to reduce the complexity or the component parts of an algebraic equation which makes it simpler and easier to understand.

The given equation is: 2(x + 7)(x² - 3x - 6).

Simplifying the given algebraic equation:

⇒ 2 (x + 7) (x² - 3x - 6)

⇒ (2x + 14) (x² - 3x - 6)

⇒ 2x³ + 14x² - 6x² - 42x - 12x - 84

⇒ 2x³ + 8x² - 54x - 84

Answer:

So we have the two real points (3/2 , 9/2)  and (-1,2).

(Question: are you wanting to use the possible rational zero theorem? Please let me know if I didn't answer your question.)

Step-by-step explanation:

y=2x^2

y^2=x^2+6x+9

is the given system.

So my plain here is to look at y=2x^2 and just plug it into the other equation where y is.

(2x^2)^2=x^2+6x+9

(2x^2)(2x^2)=x^2+6x+9

4x^4=x^2+6x+9

I'm going to put everything on one side.

Subtract (x^2+6x+9) on both sides.

4x^4-x^2-6x-9=0

Let's see if some possible rational zeros will work.

Let' try x=-1.

4-1+6-9=3+(-3)=0.

x=-1 works.

To find the other factor of 4x^4-x^2-6x-9 given x+1 is a factor, I'm going to use synthetic division.

-1   |  4     0     -1     -6    -9

    |         -4     4     -3      9

    |________________ I put that 0 in there because we are missing x^3

        4    -4     3     -9      0

The the other factor is 4x^3-4x^2+3x-9.

1 is obviously not going to make that 0.

Plug in -3 it gives you 4(-3)^3-4(-3)^2+3(-3)-9=-162 (not 0)

Plug in 3 gives you 4(-3)^3-4(-3)^2+3(-3)-9=72 (not 0)

Plug in 3/2 gives you 4(3/2)^2-4(3/2)^2+3(3/2)-9=0 so x=3/2 works as a solution.

Now let's find another factor

3/2  |     4       -4           3        -9

      |                6           3          9

      |________________________

            4          2           6        0

So we have 4x^2+2x+6=0.

The discriminant is b^2-4ac which in this case is (2)^2-4(4)(6). Simplifying this gives us (2)^2-4(4)(6)=4-16(6)=4-96=-92.  This is negative number which means the other 2 solutions are complex (not real).

So the other real solutions that satisfy the system is for x=3/2 or x=-1.

Since y=2x^2 then for x=3/2 we have y=2(3/2)^2=2(9/4)=9/2 and for x=-1 we have y=2(1)^2=2.

So we have the two real points (3/2 , 9/2)  and (-1,2)