What is (f+g)(x)? f(x)= – x2+7x g(x)=3x Write your answer as a polynomial or a rational function in simplest form.

If the frame has a 400 square inch surface area, its breadth is 5 inches.

Let's call the width of the frame w.

The total width of the painting and the frame is 24 inches + 2w inches since there is a frame of width w on both sides of the painting.

Similarly, the total height of the painting and the frame is 18 inches + 2w inches, since there is a frame of width w on both the top and bottom of the painting.

The area of the frame is the difference between the area of the rectangular frame and the area of the painting. The area of the rectangular frame is (24 inches + 2w inches) * (18 inches + 2w inches) = 432 inches + 72w inches + 36w inches + 4w^2 inches. The area of the painting is 24 inches * 18 inches = 432 inches.

Therefore, the area of the frame is 432 inches + 72w inches + 36w inches + 4w^2 inches - 432 inches

= 72w inches + 36w inches + [tex]4w^2[/tex] inches = 400 inches.

Combining like terms, we get [tex]4w^2[/tex] inches + 108w inches = 400 inches.

Dividing both sides by 4, we get [tex]4w^2[/tex] inches + 27w inches = 100 inches.

We can use the quadratic formula to solve this equation in the quadratic form:

w = [tex]\frac{(-b +/- sqrt(b^2 - 4ac))}{2a}[/tex]

= [tex]\frac{(-27 +/- sqrt(27^2 - 4 * 1 * 100))}{2*1} \\[/tex]

= [tex]\frac{(-27 +/- sqrt(729 - 400))}{2}[/tex]

= [tex]\frac{(-27 +/- sqrt(329)) }{2}[/tex]

= [tex]\frac{(-27 +/- sqrt(289))}{2}[/tex]

The solutions are w =[tex]\frac{ (-27 + 17)}{2}[/tex]  = -5 inches and w = [tex]\frac{(-27 - 17)}{2}[/tex] = -22 inches.

Since the width of the frame must be positive, the correct solution is w = -5 inches. The frame is five inches wide.

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It is a straightforward application of the pigeonhole principle to demonstrate that there must be at least one duplication in nums. Here, each possible number in nums is a "pigeon," and each specific place where a pigeon can occur is a "pigeonhole."

How to determine whether there is at least one duplicate number in Nums:

Simple Method: The goal is to use nested loops and determine whether or not each element appears in the array more than once for each element.

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The graph of y=3x is a straight line, it can be graphed by taking two points on the line.

What is a graph?

A diagram that depicts the relationship between two variables, usually measured along opposite axes in a pair, is used.

The equation we have is y=3x.

Since it is a linear equation, the graph will be a straight line.

A Line can be plotted with two points.

The two points on line y=3x are (0,0) and (1,3)

Because when x=0, y=0 and  when x=1, y=3.

We have to plot the two points on the graph and join the points and extend the line segment.

This way we can graph y=3x.

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Answer:

Scalene, isosceles and equilateral triangle are the types of triangles

Step-by-step explanation:

They  differ from each other based on their side-length. If all the three sides are different in length, then its scalene triangle. If any two sides are equal in length, then it is an isosceles triangle.

b. The probability of drawing a red card, placing it back in the deck, and drawing another red card is approximately 22%.

This is because there are 26 red cards in a standard deck of 52 cards, out of which half (13) are red. So, when Eric draws a red card and puts it back in the deck, the probability of drawing another red card is 13/52, which is approximately 25%.

However, since the deck is shuffled after the first draw, this probability is slightly reduced to 22%. Thus, the probability of drawing a red card, placing it back in the deck, and drawing another red card is approximately 22%.

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The value of x³ 3y²x² is 27y² when x is 3. This is because x³ is equal to 27, and multiplying it by 3y² and x² gives 27y².

The value of x³ 3y²x² when x is 3 can be calculated by first understanding how x³ is equal to 27. This is because when something is cubed, the value is three times itself. For example, if x is 2, then x³ is 2 x 2 x 2, which is 8. If x is 3, then x³ is 3 x 3 x 3, which is 27. Once this is understood, the value of x³ 3y²x² when x is 3 can be calculated by multiplying 27 (the value of x³ when x is 3) by 3y² and x². In this case, x² is also equal to 3, so the equation becomes 27 x 3y² x 3, which is equal to 27y². This means that the value of x³ 3y²x² when x is 3 is 27y².

x³ = 3 x 3 x 3 = 27

x³ 3y²x² = 27 x 3y² x 3 = 27y²

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Step 1: Given the functions f(x) and g(x) , compute the limit limx→∞f(x)g(x) lim x → ∞ f ( x ) g ( x ) .

Step 2: If the limit in Step 1 is a finite constant a≠0 a ≠ 0 , then the growth rate of f(x) is a times the growth rate of g(x) at sufficiently large x .

Now, According to the question:

Steps on How to Compare the Rates of Change of Two Functions Using Limits

Step 1: Given the functions f(x) and g(x), compute the limit

limx→∞f(x)/g(x).

Step 2: If the limit in Step 1 is a finite constant a≠0, then the growth rate of f(x) is a times the growth rate of g(x) at sufficiently large x. If the limit in Step 1 is ∞, then the growth rate of f(x) is greater than the growth rate of g(x). If the limit in Step 1 is 0, then the growth rate of g(x) is greater than the growth rate of f(x).

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The percent error of Alex’s estimate will be 10.5% which is the correct answer would be an option (D).

To find the percent error of Alex's estimate, we need to first calculate the difference between the estimated value and the actual value.

In this case, the difference is 42 mL - 38 mL = 4 mL.

Now we can divide the difference by the actual value and multiply by 100% to find the percent error.

As per the given question, the percent error is :

⇒ (4 mL / 38 mL) x 100%

⇒ 10.5%.

Therefore, the answer is 10.5%.

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Answer:

[tex]\displaystyle{\theta=\dfrac{5\pi}{4}\, , \, \dfrac{9\pi}{4}}[/tex]

Step-by-step explanation:

Given the equation [tex]\displaystyle{2\cos \theta = 2\sin \theta}[/tex]. We have to divide both sides by 2 which gives us [tex]\displaystyle{\cos \theta = \sin \theta}[/tex]. Then divide both sides by [tex]\displaystyle \cos \theta[/tex] which gives us [tex]\displaystyle{1=\dfrac{\sin \theta}{\cos \theta}}[/tex].

We know that:

[tex]\displaystyle{\dfrac{\sin \theta}{\cos \theta} = \tan \theta}[/tex]

So we can rewrite the equation as [tex]\displaystyle{1=\tan \theta}[/tex].

We know that [tex]\displaystyle{\tan \theta = 1}[/tex] when [tex]\displaystyle{\theta = \dfrac{\pi}{4}+\pi k}[/tex] for [tex]\displaystyle{k \in I}[/tex] (k is any integer). However, the equation is given with the interval [tex]\displaystyle{\pi \leq \theta \leq 3\pi}[/tex]. Therefore, we have to substitute k-values that satisfy the interval.

It appears that only k = 1, 2 which gives us [tex]\displaystyle{\theta_1 = \dfrac{\pi}{4}+\pi}[/tex] and [tex]\displaystyle{\theta_2 = \dfrac{\pi}{4}+2\pi}[/tex]can be used since both satisfies both values and interval.

Simplifying both solutions:

[tex]\displaystyle{\theta_1 = \dfrac{\pi}{4}+\dfrac{4\pi}{4} \, , \, \theta_2=\dfrac{\pi}{4}+\dfrac{8\pi}{4}}\\\\\displaystyle{\theta=\dfrac{5\pi}{4}\, , \, \dfrac{9\pi}{4}}[/tex]

Answer:

-28

Step-by-step explanation:

The fact that four identical unit objects arranged in a square formation would produce a square with length 2, rather than length -2, provides the logical basis for the fact that the "principal" square root of 4 is 2 rather than -2.

What is the square root?

The factor that can be multiplied by itself to obtain a given number is known as the square root. The square root is represented by The opposite of squaring a number is finding its square root.

According to the given question:

The value of root 4 exactly corresponds to 2. However, we can say that there are always two roots for any given number, and that the roots can be either positive or negative. Consequently, the value of root 4 is 2, or +2 and -2 (positive 2 and negative 2).

Hence ,the square root of 4 only 2 and not - 2.

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Add 5 to 2 = 2 will make 7 = 7 a true statement and adding 5 to 2 ≠ 3 to make 7 ≠ 8 will make a true statement.

The number system is a way to represent or express numbers.

A decimal number is a very common number that we use frequently.

Any of the multiple sets of symbols and the guidelines for utilizing them to represent numbers are included in the Number System.

(1)

As per the given,

2 = 2

Add 5 on both sides of the above equation,

2 + 5 = 2 + 5

7 = 7 (True)

(2)

As per the given

2 ≠ 3

Add 5 on both sides of the above equation,

2 + 5 ≠ 3 + 5

7 ≠ 8 (True)

Hence "It is correct that if you add 5 to 2 = 2, you will get 7 = 7, and if you add 5 to 2 ≠ 3, you will get 7 ≠ 8".

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The measure of the section HK of the parallelogram will be 13 inches.

A parallelogram is a straightforward quadrilateral with two parallel sides. A parallelogram's opposite or facing sides are of equal length, and its opposite angles are of equal measure.

A diagonal in geometry is a line segment that connects two vertices of a polygon or polyhedron that are not on the same edge. Informally, any sloping line is referred to as a diagonal.

The length of the HK will be calculated as,

HK = √ ( 12² + 5²)

HK = √ ( 144 + 25 )

HK = √ (169 )

HK = 13 inches

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Congruent figures are produced through rigid transformations. Congruent figures may always be connected to stiff transformations even though you could conceive of them as forms that "look precisely the same."

Separation between each pair of points of a geometric object is maintained during exact modification of the object. In other words, exact transformations preserve the size and shape of objects. For example, a triangle rigid transformation preserves the triangle's angles and lengths. Imagine the distortion of a rigid shape similar to that of cardboard. The shape does not change even if it is slid left and right, rotated, and mirrored.

It was the opposite when I made a mold with putty. After that, you can twist and stretch things. However, these conversions are not exact. Stretching the shape changes the distance between each pair of points.

We may produce congruent shapes by applying rotations, reflections, and translations—three types of transformations. In reality, all congruent shape pairings may be matched to one another by combining one or more of these three transformations.

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The return rate is unreasonably low. This is so since the investment is $26,245 and the account balance is $26,292 according to the quarterly statement. It is obvious that the investment has increased. Furthermore, the actual rate of return is 0.18% (($26,292-$26,244)/ $26,292)*100).

What does return on investment mean?

The percentage change in an investment's value is represented by the annual rate of return. For instance, if you estimate a 10% annual rate of return, you're anticipating that your investment's value would rise by 10% annually.

What exactly does compounding mean?

Compounding is the method through which interest is added to both the principle balance already in place and the interest that has already been paid. Thus, compounding can be thought of as interest on interest, with the result that returns on interest are magnified over time, or the so-called "magic of compounding."

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The actual distance between Eagle Lake and swallow lake is 9/1400 km.

What is distance?

Distance is the measure of the amount of space between two points or objects. It can be measured in a variety of ways, including miles, kilometers, feet, and meters. Distance is an important concept in physics, mathematics, and many other scientific fields. In addition, distance is a key factor when considering the speed of objects or events. Distance can also be used to measure the time it takes for an object or event to travel from one point to another. Distance is a fundamental concept in understanding the physical universe, and it is essential for navigation, navigation systems, and communication.

The scale of the map is given as,

                          14 m = 1 Km

We know that, 1 m = 100 cm

So, 14 × 100 cm = 1 Km

      1 cm = 1/1400 Km

So, 9 cm = 9 × 1/1400 km

               = 9/1400 km.

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The nature of the roots of the given quadratic equation 3x²+ 4x - 5 = 0 are real and distinct.

As given in the question,

Given quadratic equation is written as:

3x²+ 4x - 5 = 0

Standard form of quadratic equation is :

ax² + bx + c = 0

And nature of the roots depends on the value of discriminant 'D' .

D = b² - 4ac

here b = 4 , a = 3 , and c = -5

Discriminant 'D' = 4² - 4(3)(-5)

⇒ D = 16 + 60

⇒ D = 76 > 0

When the value of discriminant 'D' > 0 then nature of the roots are real and distinct.

Therefore, the nature of the roots of the given equation is real and distinct.

The above question is incomplete ,the complete question is :

How do you describe the nature of the roots of 3x²+ 4x - 5 = 0?

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Yes, 4096 is a perfect cube. The number whose cube is 4096 is 16. To find the cube root of a number, you can use the following formula:

cube root of a number = ∛a

For example, to find the cube root of 4096, you would calculate:

∛4096 = 16

This means that 16 is the number whose cube is 4096. You can verify this by calculating 16 x 16 x 16, which is equal to 4096.

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We can solve for x and y in two equations by using the Elimination method for the simultaneous linear equations.

The Elimination approach results in the reduction of one problem to a single-variable equation when two simultaneous linear equations are solved. Following this, the answer is the same as when one line was vertical or parallel. The Gaussian elimination method is the name given to this approach.

We can understand the above method through an example:

Equation 1:     4x + 6y = 8

Equation 2:     3x + 4y = 5

Step 1: In order to ensure that the two equations have the same leading coefficient, multiply each equation by a reasonable value. Equation 1 may be easily solved by multiplying it by 3, the x coefficient from Equation 2, and Equation 2 by 4, the x coefficient from Equation 1.

Therefore the equations become,

12x + 18y = 24

12x + 16y = 10

Step 2: Subtract equation 2 from equation 1

We get 2y = 14

From the above equation, we get the value of y as 7

Step 3: We will substitute the value of y in one of the initial equations

Substituting the value of y in equation (1) 4x + 6y = 8

4x + 6(7) = 8

4x + 42 = 8

4x = 34

x = [tex]\frac{34}{4}[/tex]

x = [tex]\frac{17}{2}[/tex]

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The ASA criterion might be the most appropriate, as it states that two triangles are congruent if two angles and the included side are congruent.

To use a congruence criterion in the given situation, you will need to determine which criterion is most appropriate based on the information provided. There are several congruence criteria that can be used to determine whether two triangles are congruent, including the Side-Side-Side (SSS) criterion, the Side-Angle-Side (SAS) criterion, the Angle-Side-Angle (ASA) criterion, and the Angle-Angle-Side (AAS) criterion.

In this case, the ASA criterion might be the most appropriate, as it states that two triangles are congruent if two angles and the included side are congruent. Given that ∠A=∠C=90∘ and AE=CB, you can use the ASA criterion to conclude that triangles AEB and CED are congruent.

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The question is -

Which congruence theorem can be used to prove EB = BD, AE = CB, ∠ A = ∠ C = 90°?

Therefore , The HCF is the divisor if the residual is 0, otherwise, apply the lemma to b and r and continue dividing until the remaining is 0.

It is possible to evaluate HCF using more than one number. It is the most effective divisor for any couple of integer that may evenly or completely divide the inputted numbers. For instance, 15 is the highest common factor between 60 and 75 since it is the largest number that divides both 60 and 75 exactly.

Here,

To calculate the HCF of the supplied numbers, many HCF formulas are utilized. See what they are now:

Product of the Numbers = LCM * HCF If P and Q are two distinct numbers, then P Q is the result of LCM (P & Q) HCF (P & Q).

Coprime numbers' HCF is always 1. LCM of Co-Prime Numbers, thus, equals the product of the numbers.

Division of Euclid To determine the HCF of two numbers, apply the lemma. Obtain the two numbers a and b such that a > b. Locate a/b , for example. The HCF is the divisor if the residual is 0, otherwise, apply the lemma to b and r and continue dividing until the remaining is 0.

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All equilateral triangles are NOT congruent.

Only those equilateral triangles whose sides have the same length will be congruent.

Those whose sides have different lengths will not be congruent.

when the discriminant is negative, that is, less than 0, then the roots are imaginary.

The discriminant in a quadratic equation is equal to:

b^2 - 4*a*c, where b is the coefficient of the degree 1 variable, a is the coefficient of the degree 2 variable and c is the constant term. There can be 3 possibilities regarding the discriminant of a quadratic equation:

Case 1: When b^2 - 4*a*c = 0

In this case, the roots of the equation are equal and real.

Case 2: When b^2 - 4*a*c > 0

In this case, the roots of the equation are real.

Case 3: When b^2 - 4*a*c < 0

In this case, the roots of the equation are imaginary.

So, when the discriminant is negative, that is, less than 0, then the roots are imaginary.

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Yes, x +3 is a factor of polynomial X³ +3x² -4x -12

Polynomial:

In mathematics, a polynomial is an expression which includes indeterminates (also called variables) and coefficients, that entails handiest the operations of addition, subtraction, multiplication, and effective-integer powers of variables. An instance of a polynomial of a single indeterminate x is x² − 4x + 7. An example with 3 indeterminates is

                  x³ + 2xyz² − yz + 1.

Polynomials appear in lots of areas of arithmetic and technological know-how. For example, they're used to form polynomial equations, which encode a wide range of issues, from elementary word troubles to complicated clinical issues; they may be used to outline polynomial functions, which seem in settings ranging from basic chemistry and physics to economics and social science; they're utilized in calculus and numerical analysis to approximate other features. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which can be primary concepts in algebra and algebraic geometry.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

(x³ + 3x²)− 4x− 12

Factor out the greatest common factor (GCF) from each group.

x²(x+3)−4(x+3)

Factor the polynomial by factoring out the greatest common factor,

x+3.(x+3)(x2−4)

Rewrite

4 as 2².

(x+3)(x²−2²)

Since both terms are perfect squares, factor using the difference of squares formula,

a²−b²=(a+ b)(a−b)

where

a = x  and b = 2.

(x+3)[(x+2)(x-2)]

Remove unnecessary parentheses.

(x+3)(x+2)(x−2)

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The simplest form of expression [tex]125\frac{2}{3}[/tex] is [tex]25[/tex].

Definition of exponent- Exponent means a quantity, describing the power to which base number is raised. In other words , Exponent refers to the number of times a number is used in a multiplication. The 4 types of exponents are: positive, negative, zero, and rational. Positive exponents tell you how many times to multiply a base by itself. Exponent is defined as the method of expressing large numbers in terms of powers.

Given :- expression: [tex]125^{\frac{2}{3} }[/tex]

Explanation:-In order to solve this expression , we use properties of exponents.

Hence , using the property - [tex]x^{m} x^{n}[/tex][tex]=x^{mn}[/tex]

and rewriting 125 as cube of 5

⇒[tex]125=5^{3}[/tex]

⇒[tex]5^({3)\frac{2}{3} }[/tex]

⇒[tex]5^{2}[/tex]

⇒[tex]25[/tex]

So, finally the simplest form of the given expression is 25.

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The solution for the system of equations is (-19,55).

As given the equations in the question

y = –3x – 2

Simplify the above

y + 3x = -2

5x + 2y = 15

Multiply y + 3x = -2 by 2 and subtracted from 5x + 2y = 15.

2y - 2y + 5x -6x = 15 + 4

-x = 19

x = -19

Putting the value of x in the equation y + 3x = -2.

y + 3 × - 19 = -2

y - 57 = -2

y = -2 + 57

y = 55

Therefore the solution for a system of equations is (-19,55).

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The equation of line in slope intercept form of line passing through the points (1,1) and (3,7) is y = 3x - 2  .

The Slope Intercept form of equation of line is denoted as y = mx + c , where m is the slope and c is y intercept .

the required line passes through the points (1,1) and (3,7) ,

So , the slope(m) is = (7-1)/(3-1) = 6/2 = 3 .

putting the value in slope intercept form  , we get ;

1 = 3(1) + c ;

1 = 3 + c ;

c = 1 - 3 ;

c = -2 .

the equation of line will become ⇒ y = 3x - 2 .

Therefore , the equation of the line in slope intercept form is y = 3x - 2 .

The given question is incomplete , the complete question is

What is the equation in the slope intercept form of a line passing through the given points (1,1) and (3,7)  ?

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Answer: The square of 2[tex]\sqrt{3[/tex] is 12

Step-by-step explanation:

squaring the 2 and [tex]\sqrt{3}[/tex] , then multiplying the results

[tex]2^{2}[/tex] =4

again [tex]\sqrt{3} ^{2}[/tex]= 3

therefore, 4x3 =12

Slope-intercept form is often expressed using the formula y = mx + b, where m denotes the line's slope and b denotes the y-value of the line's y-intercept.

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