what polynomial has roots of -6, 1, and 4 ​

The system of inequalities x + y ≤ 50 and x ≥ 20 can be graphically represented as two intersecting regions in a two-dimensional space, showing the possible combinations of stools (x) and recliners (y) the new coffee shop could have.

The subject of the question is a system of inequalities which is a common topic in high school level algebra. In this case, the system of inequalities presented is x + y ≤ 50 and x ≥ 20, where 'x' represents the number of stools and 'y' represents the number of recliners in the new coffee shop.

In order to represent this system graphically, firstly, we draw two lines that correspond to the equations x + y = 50 and x = 20. The area of intersection between the two regions defined by these lines represents the solution to the system of inequalities.

For the inequality x + y ≤ 50, we shade the area below the line because the sign is 'less than or equal to', and for x ≥ 20, we shade to the right because of the 'greater than or equal to' sign. The overlap region satisfies both inequalities and represents the possible combinations of stools and recliners the coffee shop can have according to the owner's preferences.

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Answer:

[tex]\large\boxed{\left\{\begin{array}{ccc}x\leq\dfrac{20}{a}&\text{for}\ a>0\\\\x\geq\dfrac{20}{a}&\text{for}\ a<0\end{array}\right }[/tex]

Step-by-step explanation:

[tex]ax-8\leq12\qquad\text{add 8 to both sides}\\\\ax-8+8\leq12+8\\\\ax\leq20\qquad\text{divide both sides by}\ a\neq0\\\\(1)\ \text{if}\ a>0,\ \text{then:}\\\\x\leq\dfrac{20}{a}\\\\(2)\ \text{if}\ a<0,\ \text{then you must flip the sign of inequality:}\\\\x\geq\dfrac{20}{a}[/tex]

Final answer:

To solve the inequality Ax - 8 ≤ 12 for x in terms of a, add 8 to both sides to get Ax ≤ 20, and then divide by A (assuming A > 0) to find x ≤ 20/A.

Explanation:

To solve the inequality for x in terms of a, we start with the given inequality:

Ax - 8 ≤ 12.

First, we add 8 to both sides to isolate the term with x on one side:

Ax ≤ 20

Next, assuming A is not zero, we divide both sides by A to solve for x:

x ≤ 20 / A

This is the solution to the inequality, with the understanding that it only applies if A is positive, because if A were negative, we would need to reverse the inequality sign when dividing by A.

Answer:

The correct answer option is C. [tex]y>\frac{2}{3}x+3[/tex].

Step-by-step explanation:

We are given a graph and we are to determine whether which linear  inequality is represented by the graph.

We know that the grey part on the graph represents the the region which is not included in the inequality.

Also, when x = 0, the values of y can only be less than 3.

So we choose two points on the graph and we find the slope.

For example, we take the points [tex](0,3)[/tex] and [tex](3,5)[/tex].

Slope = [tex]\frac{5-3}{3-0} =\frac{2}{3}[/tex]

which makes the equation of the line [tex]y=\frac{2}{3}x+3[/tex] and inequality [tex]y>\frac{2}{3}x+3[/tex].

The solutions to the quadratic equation are x = -1/4 and x = -5.

First, we look at the coefficient of x², which is 4 in this case. We need to find two numbers whose product is 4 times 5 (the constant term) and whose sum is the coefficient of x (12 in this case). These numbers are 1 and 20, as 1 * 20 = 20 and 1 + 20 = 21.

Next, we rewrite the middle term (12x) of the quadratic expression as the sum of these two numbers:

4x² + 1x + 20x + 5 = 0.

Now, we group the terms in pairs:

(4x² + 1x) + (20x + 5) = 0.

Next, we factor out the greatest common factor from each group:

x(4x + 1) + 5(4x + 1) = 0.

Notice that we have a common binomial factor, (4x + 1), which we can factor out:

(4x + 1)(x + 5) = 0.

Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor to zero and solve for x:

   1. 4x + 1 = 0 => 4x = -1 => x = -1/4.

   2. x + 5 = 0 => x = -5.

Answer: x = -1/4, -5.

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Answer:

x= 1 when y =4 , x= 5 when y = 8 , x=2 when y = 5.

Step-by-step explanation:

y=x+3

Through this rule we have to find out the values of x when values of y are given:

y=x+3

y = 4

Substitute the value in the rule:

4=x+3

Combine the constants:

4-3=x

x= 1 when y =4

y=x+3

y = 8

8= x+3

Combine the constants:

8-3= x

5=x

x= 5 when y = 8

y=x+3

y = 5

5=x+3

Combine the constants:

5-3=x

2=x

x=2 when y = 5....

Answer:

1/32

15/32

Step-by-step explanation:

For a fair sided coin,

Probability of heads, P(H) = 1/2

Probability of tails P(T)  = 1/2

For a coin tossed 5 times,

P( All heads)

= P(HHHHH),

= P (H) x P(H) x P(H) x P(H) x P(H)

= (1/2) x (1/2) x (1/2) x (1/2) x (1/2)

= 1/32 (Ans)

For part B, it is easier to just list the possible outcomes for

"at most 2 heads" aka "could be 1 head" or "could be 2 heads"

"One Head" Outcomes:

P(HTTTT), P(THTTT) P(TTHTT), P(TTTHT), P(TTTTH)

"2 Heads" Outcomes:

P(HHTTT), P(HTHTT), P(HTTHT), P(HTTTH), P(THHTT), P(THTHT), P(THTTH), P(TTHHT), P(TTHTH), P(TTTHH)

If we count all the possible outcomes, we get 15 possible outcomes representing "at most 2 heads)

we know that each outcome has a probability of 1/32

hence 15 outcomes for "at most 2 heads" have a probability of

(1/32) x 15  = 15/32

Answer:

x = 7/2 ±sqrt(65)/ 2

Step-by-step explanation:

x^2=7x+4

Subtract 7x from each side

x^2-7x=7x-7x+4

x^2 -7x =4

Complete the square

Take the coefficient of x and divide by 2

-7/2

Then square it

(-7/2)^2 = 49/4

Add this to each side

x^2 -7x +49/4 =4+49/4

(x-7/2)^2 = 4 +49/4

(x-7/2)^2 = 16/4 +49/4

(x-7/2)^2 =65/4

Take the square root of each side

sqrt((x-7/2)^2) =±sqrt(65/4)

x-7/2 = ±sqrt(65)/ sqrt(4)

x-7/2 = ±sqrt(65)/ 2

Add 7/2 to each side

x-7/2 +7/2=7/2 ±sqrt(65)/ 2

x = 7/2 ±sqrt(65)/ 2

The solutions to the quadratic equation x^2 = 7x + 4 are x = 3 and x = -7, found using the quadratic formula and verified by substitution into the original equation.

To solve the quadratic equation x^2 = 7x + 4, we first need to bring all terms to one side of the equation to get it into the standard form ax^2 + bx + c = 0. This gives us x^2 - 7x - 4 = 0. We can then apply the quadratic formula, which is x = (-b \/- sqrt(b^2 - 4ac)) / (2a), where a, b, and c are coefficients from the equation ax^2 + bx + c = 0.

For our equation, a = 1, b = -7, and c = -4. Substituting these values into the quadratic formula gives us two solutions, which result in x = 3 and x = -7 as the solutions to the problem. To verify these solutions, we can substitute them back into the original equation and confirm they satisfy the equation, thus proving they are correct.

Answer: So the sum of all of the measures of the interior angles of a 16-sided polygon is 2520 degrees.

The sum of the interior angles of a 16-sided polygon is 2520 degrees.

The sum of the interior angles of a 16-sided polygon, or hexadecagon, is 2520 degrees, which can be calculated using the formula (n-2) x 180 degrees.

The sum of the interior angles of any polygon can be found using the formula (n-2)  imes 180 degrees, where n is the number of sides of the polygon. A 16-sided polygon is known as a hexadecagon. Using the formula, we have:

Sum of interior angles = (16-2)  imes 180 degrees

Sum of interior angles = 14  imes 180 degrees

Sum of interior angles = 2520 degrees

Therefore, the sum of the interior angles of a 16-sided polygon is 2520 degrees.

Answer:

C.

Step-by-step explanation:

You are given 3x-4y<16 and we want to see which of the ordered pairs is a solution.

These ordered pairs are assumed to be in the form (x,y).

A. (0,-4) ?

3x-4y<16 with (x=0,y=-4)

3(0)-4(-4)<16

0+16<16

16<16 is not true so (0,-4) is not a solution of the given inequality.

B. (4,-1)?

3x-4y<16 with (x=4,y=-1)

3(4)-4(-1)<16

12+4<16

16<16 is not true so (4,-1) is not a solution of the given inequality.

C. (-3,-3)?

3x-4y<16 with (x=-3,y=-3)

3(-3)-4(-3)<16

-9+12<16

   3<16 is true so (-3,-3) is a solution to the given inequality.

D. (2,-3)?

3x-4y<16 with (x=2,y=-3)

3(2)-4(-3)<16

6+12<16

18<16 is false so (2,-3) is not a solution to the given inequality.

Answer:

D) After 270 years

Answer:

Step-by-step explanation:

Given that in 1928, f(x) = the winning jump

for women was 70.0 inches and g(x) = the winning jump for men was 86.5 inches.

Increase = 0.48% for women and 0.4%  for men

i.e. after x years  [tex]f(x) =70(1.0048)^x \\\\g(x) = 86.5(1.004)^x[/tex]

Let us find when these two values would be equal.

That is at point of intersection

Solving we get x =244 years

Hence approximately after 248 years women will exceed men.

Answer:

(72/17) hours

Step-by-step explanation:

Time Jackie would take alone , J = 8 hrs

Time Patricia would take alone , P = 9 hrs

Let the time they will take together be T

use the formula for shared unit rate

[tex]\frac{1}{T}[/tex] = [tex]\frac{1}{J}[/tex] + [tex]\frac{1}{P}[/tex]

[tex]\frac{1}{T}[/tex] = [tex]\frac{1}{8}[/tex] + [tex]\frac{1}{9}[/tex]

[tex]\frac{1}{T}[/tex] = [tex]\frac{17}{72}[/tex]

T = [tex]\frac{72}{17}[/tex] hours (or 4.24 hours)

It takes them to work together for 4 hours and 14 minutes.

A ratio is an ordered pair of numbers a and b, written as a/b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other.

Given

Jackie was to paint her living room alone. It would take 8 hours.

Her sister Patricia could do the job in 9 hours.

How long would it take them to work together?

We know the work is inversely proportional to the time. And formula we have

[tex]\rm \dfrac{1}{T_f} = \dfrac{1}{T_1} +\dfrac{1}{T_2}[/tex]

We have

[tex]\rm T_1 = 8, \ \ \ and\ \ T_2 = 9[/tex]

Then by the formula.

[tex]\rm \dfrac{1}{T_f} = \dfrac{1}{8} +\dfrac{1}{9}\\\\\rm \dfrac{1}{T_f} = \dfrac{8+9}{8*9} \\\\\rm \dfrac{1}{T_f} = \dfrac{17}{72} \\\\T_f \ = \dfrac{72}{17}\\\\T_f \ = 4.24[/tex]

Then the time 4.24 will be 4 hours and 14 minutes.

Thus, it takes them to work together for 4 hours and 14 minutes.

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Answer:

C- Negative Slope

This is because it you stated it is decreasing and it is proportional.

Negative slope best describes a condition in which a quantity decreases at a rate that is proportional to the current value of the quantity .

The slope of a line is the measure of the steepness and the direction of the line. Finding the slope of lines in a coordinate plane can help in predicting whether the lines are parallel, perpendicular, or none without actually using a compass.

Depending upon the relationship between the two variables x and y and thus the value of the gradient or slope of the line obtained.

There are 4 different types of slopes, given as,

According to the question

A quantity decreases at a rate that is proportional to the current value of the quantity

As,

Slope decreases i.e when x increases, y decreases.

and the rate at which it is decreasing is proportional to the current value of the quantity .

Therefore ,

It best describes the negative slope .

Hence, the Negative slope best describes a condition in which a quantity decreases at a rate that is proportional to the current value of the quantity .

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Answer:

y = - 3x + 3

Step-by-step explanation:

The equation of a line in slope intercept form is

y = mx + c ( m is the slope and c the y- intercept )

here m = - 3, hence

y = - 3x + c ← is the partial equation of the line

To find c substitute (- 1, 6 ) into the partial equation

6 = 3 + c ⇒ c = 6 - 3 = 3

y = - 3x + 3 ← equation of line

Answer: A

Step-by-step explanation:

FLVS Question, the answer is A !!

Answer:

3

Step-by-step explanation:

Plug in the values for a and b=     -7(-1)-2(2)

Multiply=   7-4

Subtract=   3

Hope this helps ^-^

Answer:

3

Step-by-step explanation:

We'd just substitute the value provided to us with the variable.

-7(-1) - 2(2)

-7 * -1 = 7

-2(2) = -4

7-4 = 3

Our answer is 3

Answer:

x = 4.2, LJ = 14.2

Step-by-step explanation:

When 2 chords of a circle intersect, then the product of the measures of the parts of one chord is equal to the product of the measures of the parts of the other chord, that is

10x = 6 × 7 = 42 ( divide both sides by 10 )

x = 4.2

Hence LJ = 10 + x = 10 + 4.2 = 14.2

The aquarium, which has a volume of 770 cubic feet, can hold approximately 5,760 gallons of water when taking into consideration the conversion rate of 7.48 gallons per cubic foot.

To calculate the volume of water an aquarium can hold, we need to first calculate the volume of the aquarium itself, which is determined by multiplying the length, width, and height together. In this case, we have an aquarium that measures 11 feet wide, 10 feet long and 7 feet deep, so multiplying these dimensions together gives us a volume of 770 cubic feet.

Next, we need to convert this volume into gallons. We're given the conversion rate of 7.48 gallons per cubic foot of water, so we multiply our previously obtained volume by this rate. This gives us: 770 cubic feet * 7.48 gallons per cubic foot, which equals 5,759.6 gallons.

So, the aquarium can hold approximately 5,760 gallons of water.

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Answer:

6x + 8y

Step-by-step explanation:

Distribute 2:

Note: This means to multiply 2 with the numbers inside the parentheses.

2 * 3x = 6x

2 * 4y = 8y

Our answer would be 6x +8y

Answer:

I think A and C are because they all go back to the original equation.

Step-by-step explanation:

Hope my answer has helped you and if not i'm sorry.

Answer:

c = -5

Step-by-step explanation:

Plug in -5 to c in the equation:

c² - 25 = 0

(-5)² - 25 = 0

Simplify. First, solve the power, then solve the subtraction:

(-5)² - 25 = 0

(-5 * -5) - 25 = 0

(25) - 25 = 0

0 = 0 (True)

~

Answer:

c=-5

Step-by-step explanation:

c^2-25=0

I'm going to solve this by using square root after I get the square termed by itself.

[tex]c^2-25=0[/tex]

Add 25 on both sides:

[tex]c^2=25[/tex]

Square root both sides:

[tex]c=\pm \sqrt{25}[/tex]

[tex]c=\pm 5[/tex]

Check!

[tex](5)^2-25=0 \text{ and } (-5)^2-25=0[/tex]

The diagram that represents the contrapositive of the statement "If it is an equilateral triangle, then it is an isosceles triangle" is: B. Figure B.

In Mathematics, a conditional statement is a type of statement that can be written to have both a hypothesis and conclusion. This ultimately implies that, a conditional statement has the form "if P then Q."

P → Q

Where:

P and Q represent sentences or statements.

Generally speaking, the contrapositive of a conditional statement involves interchanging the hypothesis and conclusion, and negating both hypothesis and conclusion;

~Q → ~P

In this context, the contrapositive of the given statement "If it is an equilateral triangle, then it is an isosceles triangle" can be written as follows;

"If it is not an isosceles triangle, then it is not an equilateral triangle."

Therefore, only figure correctly represent the contrapositive of the statement.

Complete Question:

Which of the diagrams below represents the contrapositive of the statement

"If it is an equilateral triangle, then it is an isosceles triangle"?

A. Figure A

B. Figure B

Answer:

jjjjjj

Step-by-step explanation:

it would be the same as before because translated means it stays the same

Answer:

6 Hours

Step-by-step explanation:

360 is half of 720, so it would take half the time to travel. half of 12 is 6

At a constant speed of 60 miles per hour, it would take 6 hours to travel 360 miles, which is a reasonable answer since the time required is halved when the distance is halved.

The question involves calculating the time it would take to travel a certain distance given a constant speed which is a basic concept in mathematics, more specifically in the topic of rates and ratios.

If you travel 720 miles in 12 hours, you are traveling at a speed of 720 miles / 12 hours = 60 miles per hour. Now, to find out how long it would take to travel 360 miles at this constant speed, you divide the distance by the speed to get the time: 360 miles / 60 miles per hour = 6 hours. So, it would take 6 hours to travel 360 miles if you maintain the same speed.

When you check if the answer is reasonable, consider if the distance is halved, the time should also be halved if the speed remains constant. Since 360 miles is half of 720 miles, and 6 hours is half of 12 hours, the answer is indeed reasonable.

Answer:

The correct option is A

Step-by-step explanation:

(x^2+x+9)+(7x^2+5)​

Open the parenthesis:

=x²+x+9+7x²+5

Now add the like terms:

=8x²+x+14

Therefore the correct option is A...

Answer:

A

Step-by-step explanation:

Elisa's distance from home (in miles) equals her walking speed (2 mph) multiplied by time spent walking (t hours).

To model Elisa's distance from home based on the time spent walking, we can use the formula for distance, which is:

[tex]\[ \text{Distance} = \text{Rate} \times \text{Time} \]\\[/tex]

Given that Elisa walks at a steady pace of 2 miles per hour, her rate (or speed) is 2 miles per hour. Let's denote this rate as [tex]\( r = 2 \)[/tex] mph.

The time Elisa spends walking can vary, so let's denote it as [tex]\( t \)[/tex] (in hours).

Now, to find Elisa's distance from home, we'll substitute the values into the formula:

[tex]\[ \text{Distance} = r \times t \]\[ \text{Distance} = 2 \times t \][/tex]

Since Elisa's distance from home is what we're interested in, this equation models her distance from home based on the time spent walking. It shows that her distance from home increases linearly with time as she walks at a steady pace.

[tex]\bf \cfrac{1+cot^2(\theta )}{1+csc(\theta )}=\cfrac{1}{sin(\theta )} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{1+cot^2(\theta )}{1+csc(\theta )}\implies \cfrac{1+\frac{cos^2(\theta )}{sin^2(\theta )}}{1+\frac{1}{sin(\theta )}}\implies \cfrac{~~\frac{sin^2(\theta )+cos^2(\theta )}{sin^2(\theta )}~~}{\frac{sin(\theta )+1}{sin(\theta )}}\implies \cfrac{~~\frac{1}{sin^2(\theta )}~~}{\frac{sin(\theta )+1}{sin(\theta )}}[/tex]

[tex]\bf \cfrac{1}{\underset{sin(\theta )}{~~\begin{matrix} sin^2(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ }}\cdot \cfrac{~~\begin{matrix} sin(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{sin(\theta )+1}\implies \cfrac{1}{sin^2(\theta )+sin(\theta )} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \cfrac{1+cot^2(\theta )}{1+csc(\theta )}\ne \cfrac{1}{sin(\theta )}~\hfill[/tex]

Step-by-step explanation:

The y-intercept is the value of the function at x = 0.

f(0) = -3(0) + 2 = 2

g(0) = -3

h(0) = 4 sin(0 + π) + 3 = 3

h(x) has the greatest y-intercept.

Answer:

The y-intercept of functions f(x), g(x) and h(x) are 2,-3 and 3 respectively. Therefore the function h(x) has the greatest y-intercept.

Step-by-step explanation:

The given function is

[tex]f(x)=-3x+2[/tex]

Substitute x=0, to find the y-intercept of the function.

[tex]f(0)=-3(0)+2[/tex]

[tex]f(0)=0+2[/tex]

[tex]f(0)=2[/tex]

The y-intercept of the function f(x) is 2.

From the given graph it is clear that the graph of g(x) intersect the y-axis at y=-3.

Therefore the y-intercept of the function g(x) is -3.

The given function is

[tex]h(x)=4\sin (2x+\pi)+3[/tex]

Substitute x=0, to find the y-intercept of the function.

[tex]h(0)=4\sin (2(0)+\pi)+3[/tex]

[tex]h(0)=4\sin (0+\pi)+3[/tex]

[tex]h(0)=4\sin (\pi)+3[/tex]

[tex]h(0)=4(0)+3[/tex]

[tex]h(0)=3[/tex]

The y-intercept of the function h(x) is 3.

The y-intercept of functions f(x), g(x) and h(x) are 2,-3 and 3 respectively. Therefore the function h(x) has the greatest y-intercept.

Answer:

The answer is B on edge

Step-by-step explanation:

The area of the cross section is equal to 90 ft²

Looking at the diagram we would see that the two dimensional solid that passed the point is a triangle.

[tex]\frac{1}{2} bh[/tex]

Where b = bas

h = height

The radius of the cone = 6

The diameter of the cone = 2*radius

= 2*6

= 12

We have to put d = b = 12

When we put the values into the area of a triangle

= [tex]\frac{1}{2} 12*15\\\\= \frac{180}{2} \\\\= 90 ft^2[/tex]

The area of the cross section is therefore 90 ft²

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Answer:

(x + 8)^2.

Step-by-step explanation:

x^2 + 16x + 64

8 + 8 = 16 and 8^2 = 64 so the factors are

(x + 8)(x + 8) or (x + 8)^2

Answer:

[tex]\large\boxed{y=\dfrac{15}{14}x+1.5}[/tex]

Step-by-step explanation:

The slope-intercept form of an equation of a line:

[tex]y=mx+b[/tex]

m - slope

b - y-intercept

------------------------------------------------------------------------

You must solve the system of equations:

[tex]\left\{\begin{array}{ccc}4x-3y+4=0&(1)\\x+2y=5&(2)\\y=mx+1.5&(3)\end{array}\right\qquad\text{substitute (3) to (1) and (2)}\\\\\left\{\begin{array}{ccc}4x-3(mx+1.5)+4=0\\x+2(mx+1.5)=5\end{array}\right\qquad\text{use the distributive property}\\\left\{\begin{array}{ccc}4x-3mx-4.5+4=0\\x+2mx+3=5&\text{subtract 3 from both sides}\end{array}\right\\\left\{\begin{array}{ccc}4x-3mx-0.5=0&\text{add 0.5 to both sides}\\x+2mx=2\end{array}\right\\\left\{\begin{array}{ccc}4x-3mx=0.5&\text{multiply both sides by 2}\\x+2mx=2&\text{multiply both sides by 3}\end{array}\righ[/tex]

[tex]\underline{+\left\{\begin{array}{ccc}8x-6mx=1\\3x+6mx=6\end{array}\right}\qquad\text{add both sides of the equations}\\.\qquad11x=7\qquad\text{divide both sides by 11}\\.\qquad x=\dfrac{7}{11}\\\\\text{Put the value of}\ x\ \text{to the second equation:}\\\\\dfrac{7}{11}+2m\left(\dfrac{7}{11}\right)=2\qquad\text{multiply both sides by 11}\\\\7+2m(7)=22\qquad\text{subtract 7 from both sides}\\\\14m=15\qquad\text{divide both sides by 14}\\\\m=\dfrac{15}{14}[/tex]

Answer:

Rosa is closer to the correct depth.  

Step-by-step explanation:

To show that two measurements are nearly equivalent, we must convert one of the measurements to the other unit.

1 m = 3.281 ft

[tex]\text{Depth } = \text{1644 ft} \times \dfrac{\text{1 m}}{\text{3.281 ft}} = \textbf{501.1 m}[/tex]

Neither is correct but Rosa is closer to the correct depth.