Kevin wants to buy an area rug for his living room. He would like the area rug to be no smaller that 48 square feet and no bigger than 80 square feet. If the length is 2 feet more than the width, what are the range of possible values for the width?

A perfect square trinomial can be factored using the pattern x²+2ab+b²=(x+a)² or x²-2ab+b²=(x-a)². The term 'a' is the square root of the last term, and 'b' is half of the second term's coefficient.

In mathematics, a perfect square trinomial is a type of second-order polynomial or quadratic function with a specific form. It is used in the factorization process, particularly when solving quadratic equations of the form ax² + bx + c.

A perfect square trinomial is a trinomial in the form x²+2ab+b² or x²-2ab+b². To factor such a trinomial, the pattern (x+a)² or (x-a)² is used where 'x' is the variable, 'a' is the square root of the third term, and 'b' is half of the coefficient of the second term. Hence, x²+2ab+b² factors to (x+a)², whereas x²-2ab+b² factors to (x-a)².

Take for example, the trinomial x²+6x+9. Here, 'a' is 3 (as √9=3) and 'b' is 3 (as half of 6 is 3). Both 'a' and 'b' are equal so we know the trinomial is perfect square and it factors to (x+3)².

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The factor a perfect square trinomial, use the pattern [tex](a+b)^2[/tex] = [tex]a^2[/tex]  +2ab+  [tex]b ^{2}[/tex] and simplify .

The pattern for a perfect square trinomial is a helpful algebraic expression that represents the square of a binomial.

It takes the form [tex](a+b)^2[/tex] = [tex]a^2[/tex] +2ab+[tex]b ^{2}[/tex]  

where a and b are variables. When faced with a trinomial that fits this pattern, you can use it to factor the trinomial efficiently.

To factor a perfect square trinomial, compare it with the pattern. If the trinomial can be expressed in the form [tex](a+b)^2[/tex] =[tex]a^2[/tex]+2ab+ [tex]b ^{2}[/tex] , then it is a perfect square trinomial.

Identify [tex]a^2[/tex]  ,2ab and [tex]b ^{2}[/tex] terms in the trinomial.

Once identified, you can write the factored form as [tex](a+b)^2[/tex].  This means the trinomial can be factored as  [tex](a+b)^2[/tex] where a is the square root of the first term, and b is the square root of the last term.

This process is essentially reverse-engineering the expansion of the perfect square trinomial pattern.

Factoring using the perfect square trinomial pattern can save time compared to other methods, making it a valuable tool in algebraic manipulations and simplifying expressions.

Answer:

6 which is none of your answers.

Are you sure your function is right?

Is value to plug in x=-8?

Step-by-step explanation:

To find the value of the function at x=-8, you replace x with -8 in the function.

g(x)=3(x+10)

g(-8)=3(-8+10)

g(-8)=3(2)

g(-8)=6

g(-8) = 6.

The value of g(x) = 3(x + 10) when x = -8 is:

Substituting x = -8 in the function g(x)

g(-8) = 3 (-8+ 10)

Solving the operation in the  parenthesis

g(-8) = 3 (-8 + 10)

g(-8) = 3 (2)

Solving the multiplication

g(-8) = 6

Answer:

-6

Step-by-step explanation:

The average rate of a function f(x) on the interval from x=a to x=b is [tex]\frac{f(b)-f(a)}{b-a}[/tex].

So in the problem you have from [tex]x=2[/tex] to [tex]x=4[/tex].

The average rate of the function from x=2 to x=4 is

[tex]\frac{f(4)-f(2)}{4-2}=\frac{f(4)-f(2)}{2}[/tex].

Now we need to find f(4) and f(2).

f(4) means what is the y-coordinate that corresponds to x=4 on the curve.

f(4)=-15 since the ordered pair at x=4 is (4,-15).

f(2) means what is the y-coordinate that corresponds to x=2 on the curve.

f(2)=-3 since the ordered pair at x=2 is (2,-3).

So let's plug in those values:

[tex]\frac{f(4)-f(2)}{4-2}=\frac{-15-(-3))}{2}[/tex].

Now we just simplify:

[tex]\frac{-15+3}{2}[/tex]

[tex]\frac{-12}{2}[/tex]

[tex]-6[/tex]

Answer:

3.2

Step-by-step explanation:

If there are 2 secont lines intersecting inside a circle like the picture shown, then the theorem tells us that "the product of 2 segments of one secant line is equal to the product of 2 segments of other secant line"

Thus, we can say that:

x * 10 = 8 * 4

10x = 32

x = 32/10

x = 3.2

Answer:

[tex]\frac{(x--3)^2}{49} -\frac{(y-6)^2}{32}=1[/tex]

Step-by-step explanation:

The standard equation of a horizontal hyperbola with center (h,k) is

[tex]\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1[/tex]

The given hyperbola has vertices at (–10, 6) and (4, 6).

The length of its major axis is [tex]2a=|4--10|[/tex].

[tex]\implies 2a=|14|[/tex]

[tex]\implies 2a=14[/tex]

[tex]\implies a=7[/tex]

The center is the midpoint of the vertices (–10, 6) and (4, 6).

The center is [tex](\frac{-10+4}{2},\frac{6+6}{2}=(-3,6)[/tex]

We need to use the relation [tex]a^2+b^2=c^2[/tex] to find [tex]b^2[/tex].

The c-value is the distance from the center (-3,6) to one of the foci (6,6)

[tex]c=|6--3|=9[/tex]

[tex]\implies 7^2+b^2=9^2[/tex]

[tex]\implies b^2=9^2-7^2[/tex]

[tex]\implies b^2=81-49[/tex]

[tex]\implies b^2=32[/tex]

We substitute these values into the standard equation of the hyperbola to obtain:

[tex]\frac{(x--3)^2}{7^2} - \frac{(y-6)^2}{32}=1[/tex]

[tex]\frac{(x+3)^2}{49} -\frac{(y-6)^2}{32}=1[/tex]

Answer:

x = - 5, x = - 1

Step-by-step explanation:

To find the x- intercepts let f(x) = 0, that is

x² + 6x + 5 = 0 ← in standard form

Consider the factors of the constant term (+ 5) which sum to give the coefficient of the x- term ( + 6)

The factors are + 5 and + 1, since

5 × 1 = 5 and 5 + 1 = + 6, hence

(x + 5)(x + 1) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 5 = 0 ⇒ x = - 5 ⇒ (- 5, 0 )

x + 1 = 0 ⇒ x = - 1 ⇒ (- 1, 0)

Answer:

-5 -1

Step-by-step explanation:

on edge

Answer:

Option B. [tex]YV=5\ units[/tex]

Step-by-step explanation:

we know that

The Intersecting Secant-Tangent Theorem, states that If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment

so

In this problem

[tex]WZ^{2}=WV*WY[/tex]

substitute and solve for WV

[tex]6^{2}=WV*4[/tex]

[tex]WV=36/4=9\ units[/tex]

we have that

[tex]WV=WY+YV[/tex]

substitute

[tex]9=4+YV[/tex]

[tex]YV=9-4=5\ units[/tex]

[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{0}~,~\stackrel{y_1}{10})\qquad (\stackrel{x_2}{-9}~,~\stackrel{y_2}{1})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(-9-0)^2+(1-10)^2}\implies d=\sqrt{(-9)^2+(-9)^2} \\\\\\ d=\sqrt{81+81}\implies d=\sqrt{162}\implies d\approx 12.73[/tex]

bearing in mind that standard form for a linear equation means

• all coefficients must be integers, no fractions

• only the constant on the right-hand-side

• all variables on the left-hand-side, sorted

• "x" must not have a negative coefficient

[tex]\bf (\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{1}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{1-2}{2-(-3)}\implies \cfrac{1-2}{2+3}\implies -\cfrac{1}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-2=-\cfrac{1}{5}[x-(-3)]\implies y-2=-\cfrac{1}{5}(x+3)[/tex]

[tex]\bf y-2=-\cfrac{1}{5}x-\cfrac{3}{5}\implies y=-\cfrac{1}{5}x-\cfrac{3}{5}+2\implies y=-\cfrac{1}{5}x+\cfrac{7}{5} \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{5}}{5(y)=5\left( -\cfrac{1}{5}x+\cfrac{7}{5} \right)}\implies 5y=-x+7\implies \blacktriangleright x+5y=7 \blacktriangleleft[/tex]

The equation of the line passing through points (-3,2) and (2,15) is calculated using the point-slope formula after calculating the slope. The derived equation is y = 2.6x + 7.8.

The question is asking to find the equation of the line that passes through two given points: (-3,2) and (2,15). To tackle this, firstly we need to calculate the slope (m) of the line, which is given by the formula: (y2 - y1) / (x2 - x1). Secondly, we apply the point-slope formula, y - y1 = m(x - x1), where (x1, y1) can be either point.

Now, calculating the slope using the given points, we find (15-2) / (2- (-3)) equals to 13/5 or 2.6. Next, we use the point-slope formula to write the equation. If using point (-3,2) we find y - 2 = 2.6 (x - (-3)). Thus, the equation for the line that passes through (-3,2) and (2,15) is y = 2.6x + 7.8 after simplifying the equation.

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Answer:

[tex]x = \frac{8}{23} \: \: \: \\ y = \frac{292}{23} [/tex]

Answer:

-5, 2

Step-by-step explanation:

Answer:

Road A

Step-by-step explanation:

The slope of road A is:

154 / 2200 = 0.07

The slope of road B is:

153 / 3400 = 0.045

Road A has the larger slope, and is therefore steeper.

The steepness of a road is determined by the ratio of the vertical climb to the horizontal distance. By calculating this ratio for both Road A and Road B, we find that Road A, with a ratio of 0.07, is steeper than Road B, which has a ratio of 0.045.

The steepness of a road is determined by the ratio of the vertical climb to the horizontal distance. This is equivalent to finding the slope in mathematics. We can calculate this for both Road A and Road B:

For Road A, the slope can be calculated as rise over run, or 154 feet over 2200 feet, yielding approximately 0.07. For Road B, applying the same formula gives 153 feet over 3400 feet, resulting in a slope of about 0.045.

Comparing these two results, it is clear that Road A, with a slope of 0.07, is steeper than Road B, which has a slope of 0.045.

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Ration expressions cause excluded values wherever the denominator equals zero.

So, for any expression like

[tex]h(x)=\dfrac{f(x)}{g(x)}[/tex]

-5 is an excluded value if [tex]g(-5)=0[/tex]

For example, the simplest one would be

[tex]h(x) = \dfrac{1}{x+5}[/tex]

In fact, if you try to evaluate this function at -5, you'd have

[tex]h(-5)=\dfrac{1}{5+5}=\dfrac{1}{0}[/tex]

which is undefined, and thus you can't evaluate the function, and thus -5 is an excluded value.

Answer:

6/x+5

Step-by-step explanation:

Answer:

Step-by-step explanation:

There is not possible solution in the real number system or the complex field.

Oddly, before I answer your question, I would point out that the equation given is a perfectly legitimate equation in some computer languages. It has the meaning of

But that is not what you are being asked about.

The blank you could put in 4

so 4 = 4 - 5

4 = - 1

The second blank will give you the result of 4 = - 1 which can't be true in a million years.

Answer:

x=5

Step-by-step explanation:

If the equation is:

3(x+1)=7(x-2) -3

Solution:

Multiply (x+1) by 3 and (x-2) by 7

3x+3 = 7x-14 -3

3x+3 = 7x -17

Now combine the like terms:

3+17 = 7x-3x

20 = 4x

Now divide both the terms by 4

20/4 = 4x/4

5 = x

You can write it as x=5

Thus the value of x is 5 ....

Answer:

Option C is correct.

Step-by-step explanation:

The rate of change can be found by finding the slope of given ordered pairs.

y₂=12, y₁=7,x₂=1,x₁=0

rate of change = y₂-y₁/x₂-x₁

rate of change = 12-7/1-0

rate of change = 5/1 = 5

Now considering next two points,

y₂=17, y₁=7,x₂=2,x₁=1

rate of change = y₂-y₁/x₂-x₁

rate of change = 17-12/2-1

rate of change = 5/1 = 5

So, the rate of change for this set of ordered pairs is 5.

Option C is correct.

Answer:

= 52°

Step-by-step explanation:

The obtuse angle at O is twice the angle made at the circumference /by the same segment b.

=52×2=104°

The base angles that are made by isosceles triangle that has the apex with angle 104° equal to (180-104)/2=38°

The radii of a circle are equal and meet tangents at right angles.

Angle a= 90-38= 52°

Answer:

The probability that the proportion of Grammy award winners will be less than 47% is 0.6915

Step-by-step explanation:

* Lets explain how to solve the problem

- Suppose 46% of American singers are Grammy award winners.

- In a random sample of size 622 is selected

- We want to find the probability that the proportion of Grammy award

 winners will be less than 47%

* At first we must to calculate z

∵ [tex]z=\frac{P^{'}-P}{\sqrt{\frac{P(1-P)}{n}}}[/tex], where

# P' is the sample proportion

# n is the sample size

# P is probability of success

∵ The sample proportion is 47% = 47/100 = 0.47

∴ P' = 0.47

∵ The sample size is 622

∴ n = 622

∵ The probability of success is 46% = 46/100 = 0.46

∴ P = 0.46

∴ [tex]z=\frac{0.47-0.46}{\sqrt{\frac{0.46(1-0.46)}{622}}}=0.5004[/tex]

- P(P' < 0.47) = P(z < 0.5004)

∵ P(z < 0.5004) = 0.6915

∴ P(P' < 0.47) = 0.6915

* The probability that the proportion of Grammy award winners will

 be less than 47% is 0.6915

Answer:

a,b

Step-by-step explanation:

in the rectangle when bisected

if mid point is taken as O

BO=OD

AO=OC

X=2a

mid point of X=2a/2=a

Y=2b

y mid point = b

The midpoint of BD will be ( a,b ) so option (C) will be correct.

A line section that can connect two places is referred to as a segment.

In other terms, a line segment is merely a section of a larger, straight line that extends indefinitely in both directions.

The line is here! It extends endlessly in both directions and has no beginning or conclusion.

The midpoint of a line associated with two coordinates is given by

(x₁ + x₂)/2 and (y₁ + y₂)/2

Given that B (0,2b) D (2a,0)

So midpoint

(0 + 2a)/2 and (2b + 0)/2

⇒ (a,b) so the midpoint will be this.

For more about line segment

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Answer:

y-3=2(x-1)

Step-by-step explanation:

So it looks like your line goes through (2,5) and (1,3) based on what you have said.

Looking at 5 to 3, that is down 2.

Locking at 2 to 1, that is left 1.

So the slope is -2/-1 =2/1=2.

Plug in the (x1,y1)=(1,3) and slope=m=2.

So using point slope form we have y-3=2(x-1).

Answer:

Step 3

Step-by-step explanation:

Just did the test

Answer:

Step-by-step explanation:

[tex]3(x-4)-5=x-3\qquad\text{use the distributive property}\\3x-12-5=x-3\\3x-17=x-3\qquad\text{add 17 to both sides}\\3x-17+17=x-3+17\\3x=x+14\qquad\text{subtract}\ x\ \text{from both sides}\\3x-x=x-x+14\\2x=14\qquad\text{divide both sides by 2}\\\dfrac{2x}{2}=\dfrac{14}{2}\\\\x=7[/tex]

Answer:

x^2+11x+10

or

(x+1)(x+10) since you can factor x^2+11x+10

Step-by-step explanation:

Let's do synthetic division.

We are dividing by x+2, so -2 will be on the outside. Like this:

-2 |         1          13          32         20

   |                     -2        -22         -20

   |___________________________

             1          11           10            0

The remainder is 0, so (x+2) is indeed a factor of x^3+13x^2+32x+20.

The other factor we found by doing this is (x^2+11x+10).

You can find more factors by factoring x^2+11x+10.

Two numbers that multiply to be 10 and add to be 11 is 10 and 1 so the factored form of x^2+11x+10 is (x+10)(x+1).

First, to get applied by the fraction rule: [tex]\displaystyle\frac{3^4}{1}[/tex], then, the applied rule [tex]\displaystyle\frac{a}{1}=a[/tex]. Finally, simplify to find the answer. [tex]\displaystyle3^4=3*3*3*3=81[/tex], the correct answer is 81. I hope this will help you. Have a wonderful day!

[tex]\left(\dfrac{1}{3}\right)^{-4}=3^4=81[/tex]

Answer:

1001 ways

Step-by-step explanation:

Total number of people who applied for the job = 8 + 6 = 14

Number of people to be chosen = 4

This is a combination problem because the order of selection does not matter. A group selection involves the combinations. So here we have to find the combinations of 14 people taken 4 at a time. The formula for the combination is:

[tex]^{n}C_{r} = \frac{n!}{r!(n-r)!}[/tex]

Here, n is the total number of objects which is 14 in this case.

r is the number of objects to be chosen which is 4 in this case.

Using these values, we get:

[tex]^{14}C_{4}=\frac{14!}{4!(14-4)!}\\\\ = \frac{14!}{4! \times 10!}\\\\ =1001[/tex]

Thus, there are 1001 ways to select 4 applicants from 8 men and 6 women for the second interview.

Answer:

see explanation

Step-by-step explanation:

Calculate the slope m of the line using the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (- 7, 3) and (x₂, y₂ ) = (1, - 97)

m = [tex]\frac{-97-3}{1+7}[/tex] = [tex]\frac{-100}{8}[/tex] = - [tex]\frac{25}{2}[/tex]

Answer:

No correct answer.

Step-by-step explanation:

C: 2 or 3 is  1/2 the probability. The expectation is that you should get about 250 readings that are either 2 or 3.

A is not correct. It will land on an even number about 250 times. What should happen in theory is far different than what will happen when you try it. Exactly is too confining a word.

B: It will land on 1 about 1 out of every 4 times. 1/4 * 500 = 125. So B is not right.

D: Same as B. There is no answer.  

Answer:

[tex]\frac{\pi }{4}[/tex]

Step-by-step explanation:

To convert from degrees to radians

radian measure = degree measure × [tex]\frac{\pi }{180}[/tex]

given degree measure = 45°, then

radian = 45 × [tex]\frac{\pi }{180}[/tex] ( divide 45 and 180 by 45 )

           = [tex]\frac{\pi }{4}[/tex]

Answer:

π/4 radians.

Step-by-step explanation:

To convert to radians we multiply degrees by π/180.

So 45 degrees = 45 * π / 180

= π/4 radians.

Answer:

16

Step-by-step explanation:

8 / 2  * (2 + 2)

4 * (2 + 2)

4 * 4

16

Step-by-step explanation:

PEMDAS is ( Thank you google )

" PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. Given two or more operations in a single expression, the order of the letters in PEMDAS tells you what to calculate first, second, third and so on, until the calculation is complete."

First, 2 + 2 = 4

Second, 8 divided by 2 is 4.

Third, 4x4= 16

Therefore, the answer is 16.

Answer:

B.

Step-by-step explanation:

To simplify something that looks like [tex]\frac{\text{whatever}}{\sqrt{a}-\sqrt{b}}[/tex] you would multiply the top and bottom by the conjugate of the bottom. So you multiply the top and bottom for this problem I just made by:

[tex]\sqrt{a}+\sqrt{b}[/tex].

If you had  [tex]\frac{\text{whatever}}{\sqrt{a}+\sqrt{b}}[/tex], then you would multiply top and bottom the conjugate of [tex]\sqrt{a}+\sqrt{b}[/tex] which is [tex]\sqrt{a}-\sqrt{b}[/tex].

The conjugate of a+b is a-b.

These have a term for it because when you multiply them something special happens.  The middle terms cancel so you only have to really multiply the first terms and the last terms.

Let's see:

(a+b)(a-b)

I'm going to use foil:

First:  a(a)=a^2

Outer: a(-b)=-ab

Inner:  b(a)=ab

Last:    b(-b)=-b^2

--------------------------Adding.

a^2-b^2

See -ab+ab canceled so all you had to do was the "first" and "last" of foil.

This would get rid of square roots if a and b had them because they are being squared.

Anyways the conjugate of [tex]\sqrt{17}-\sqrt{2}[/tex] is

[tex]\sqrt{17}+\sqrt{2}[/tex].

This is the thing we are multiplying and top and bottom.

For this case we have the following expression:

[tex]\frac {3} {\sqrt {17} - \sqrt {2}}[/tex]

We must rationalize the expression, so we multiply by:

[tex]\frac {\sqrt {17} + \sqrt {2}} {\sqrt {17} + \sqrt {2}}[/tex]

So, we have:

[tex]\frac {3} {\sqrt {17} - \sqrt {2}} * \frac {\sqrt {17} + \sqrt {2}} {\sqrt {17} + \sqrt {2}} =\\\frac {3 (\sqrt {17} + \sqrt {2}} {17- \sqrt {17} * \sqrt {2} + \sqrt {17} * \sqrt {2} -2} =\\\frac {3 (\sqrt {17} + \sqrt {2}} {15}[/tex]

Thus, the correct option is option B.

Answer:

OPTION B