A student is asked to factor, if possible, the given expression. The student's response demonstrates which common misconception? Simplify: a2 + b2 Student's response: (a + b)(a − b)

Answers

Answer 1

Answer:

The student assumed he was asked to factor

(a^2 - b^2)

Which is equal to

= (a+b)*(a-b)    (His response)

But, he was asked to simplify

(a^2 + b^2)

Which has imaginary roots, and can be factored as:

(b +ai)(b-ai)


Related Questions

5 friends share 3 bags of trail mix equally. What fraction of a bag of trail mix does each friend get?

Answers

Answer

the answer is 6/10 or 0.6

Step-by-step explanation:

6/10 (3/5) because you have to share the amount of bags into the ratio of friends which would give you 0.6 or 6/10 simplified to 3/5

For #1 and 2, Find the lateral area and surface area of each prism.

Answers

The lateral area and surface area of the cuboid are 72 square feet and 142 square feet, respectively. The lateral area and surface area of the prism are 42 square cm and 62 square cm, respectively.

To find the lateral area and surface area of the given cuboid and prism, we need to use the formulas for each.

For the cuboid:
1. Lateral Area: The lateral area of a cuboid can be found by adding the areas of its four side faces. Since the cuboid has a length of 7 feet, a breadth of 5 feet, and a height of 3 feet, we can calculate the lateral area as follows:
- Side 1: length * height = 7 feet * 3 feet = 21 square feet
- Side 2: breadth * height = 5 feet * 3 feet = 15 square feet
- Side 3: length * height = 7 feet * 3 feet = 21 square feet
- Side 4: breadth * height = 5 feet * 3 feet = 15 square feet
The total lateral area is the sum of these four areas: 21 + 15 + 21 + 15 = 72 square feet.

2. Surface Area: The surface area of a cuboid is found by adding the areas of all six faces. In addition to the lateral area, we need to include the areas of the top and bottom faces. Using the dimensions of the cuboid, we can calculate the surface area as follows:
- Top face: length * breadth = 7 feet * 5 feet = 35 square feet
- Bottom face: length * breadth = 7 feet * 5 feet = 35 square feet
The total surface area is the sum of these six areas: 72 (lateral area) + 35 (top face) + 35 (bottom face) = 142 square feet.

Now, for the prism:
1. Lateral Area: To find the lateral area of the prism, we need to find the perimeter of the base and multiply it by the height of the prism. Since the prism has a base with a length of 5 cm and a breadth of 2 cm, the perimeter is: 2 * (length + breadth) = 2 * (5 cm + 2 cm) = 2 * 7 cm = 14 cm.
The lateral area is obtained by multiplying the perimeter by the height: 14 cm * 3 cm = 42 square cm.

2. Surface Area: The surface area of the prism is the sum of the lateral area and twice the area of the base. Since the base is a rectangle with a length of 5 cm and a breadth of 2 cm, its area is: length * breadth = 5 cm * 2 cm = 10 square cm.
The surface area is then calculated as: 42 (lateral area) + 2 * 10 (area of the base) = 42 + 20 = 62 square cm.

Will give brainliest if you show your work and answer completely!!!

Rick is a real estate agent and makes 7% commission on his total sales. He sells a home that worth $150,000. What is his commission?

Answers

7% into decimal is 0.07

0.07 x $150,000
= $10,500

Need the answer to #3

Answers

Answer:

  3.  P(R1|Q) = 3/19 ≈ 0.16

Step-by-step explanation:

The desired probability is the ratio of P(Q·R1) to P(Q). The probability P(Q) is not given, but there is sufficient information to find it.

  P(Q·R1) = P(Q|R1)·P(R1) = 0.40·0.15 = 0.06

  P(Q·R2) = P(Q|R2)·P(R2) = 0.20·0.55 = 0.11

  P(Q·R3) = P(Q|R3)·P(R3) = 0.70·0.30 = 0.21

Since R1 and R2 and R3 are mutually exclusive and have a joint probability of 1, this means ...

  P(Q) = P(Q·R1) +P(Q·R2) +P(Q·R3) = 0.06 +0.11 +0.21 = 0.38

Then the desired probability is ...

  P(R1|Q) = P(Q·R1)/P(Q) = 0.06/0.38

  P(R1|Q) = 3/19 ≈ 0.16

what is tan(sin^-1(x/2))

Answers

Let

[tex]\theta=\sin^{-1}\dfrac x2\implies\sin\theta=\dfrac x2[/tex]

Recall that

[tex]\cos\theta=\sqrt{1-\sin^2\theta}=\sqrt{1-\dfrac{x^2}4}[/tex]

Then

[tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}\implies\tan\left(\sin^{-1}\dfrac x2\right)=\dfrac{\frac x2}{\sqrt{1-\frac{x^2}4}}[/tex]

[tex]\implies\tan\left(\sin^{-1}\dfrac x2\right)=\dfrac x{\sqrt{4-x^2}}[/tex]

Final answer:

To find tan(sin⁻¹(x/2)), we use the relationship between the sides of a right triangle, leading to the solution tan(θ) = x/√(4 - x²).

Explanation:

The question "what is tan(sin-1(x/2))" involves understanding inverse trigonometric functions and their properties. To solve this, we can create a right-angled triangle where the angle θ has a sine value of x/2. According to the Pythagorean theorem, if one side (opposite) is x and the hypotenuse is 2, the adjacent side can be calculated as √(4 - x²). Thus, tan(θ) = opposite/adjacent = x/√(4 - x²).

1. A claim states that a specific crimes occurs in the given proportions shown in the table for October through March. Use the frequencies from randomly selected crimes to find the test statistic using a 0.01 significance level needed to test the claim.

A. 15.086
B. 9.843
C. 6.683
D. 749.500

(plus other questions in attached)

Answers

Answer:

C. 6.683

Step-by-step explanation:

How do you use the limit comparison test on this particular series?
Calculus series tests​

Answers

Compare [tex]\dfrac1{\sqrt{n^2+1}}[/tex] to [tex]\dfrac1{\sqrt{n^2}}=\dfrac1n[/tex]. Then in applying the LCT, we have

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac1{\sqrt{n^2+1}}}{\frac1n}\right|=\lim_{n\to\infty}\frac n{\sqrt{n^2+1}}=1[/tex]

Because this limit is finite, both

[tex]\displaystyle\sum_{n=1}^\infty\frac1{\sqrt{n^2+1}}[/tex]

and

[tex]\displaystyle\sum_{n=1}^\infty\frac1n[/tex]

behave the same way. The second series diverges, so

[tex]\displaystyle\sum_{n=0}^\infty\frac1{\sqrt{n^2+1}}=1+\sum_{n=1}^\infty\frac1n[/tex]

is divergent.

Final answer:

The limit comparison test is used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series.

Explanation:

The limit comparison test is used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series. The test states that if the limit of the ratio between the terms of the given series and the terms of a known convergent series is a finite positive number, then both series behave in the same way. Here is how you can use the limit comparison test step by step:

Choose a known series that converges, ideally one that is similar in nature to the given series.Take the limit of the ratio between the terms of the given series and the terms of the known convergent series.If the limit is a finite positive number, then both series converge or diverge together. If the limit is zero or infinite, the result is inconclusive.

For example, if we have the series ∑(n^2)/(2^n), we can use the limit comparison test with the series ∑(1)/(2^n). Taking the limit of the ratio (n^2)/(2^n) / (1)/(2^n) as n approaches infinity, we get:

lim(n→∞) [(n^2)/(2^n)] / [(1)/(2^n)] = lim(n→∞) (n^2)/(1) = ∞

Since the limit is infinite, the result is inconclusive. Therefore, the given series does not converge or diverge using the limit comparison test.

What is the sum of the series? ​ 5∑i=1 4i ​

Answers

Answer:

60

Step-by-step explanation:

∑[i=1,5] 4i = 4·1 + 4·2 + 4·3 + 4·4 + 4·5

= 4 + 8 + 12 + 16 + 20

= 60

Point M is located at (3, 7). If point M is translated 2 units to the left and 4 units down, what are the coordinates of M’?

Answers

It would be located at (1, 3). Hope this helps!

Answer:

1,3

Step-by-step explanation:

The figure is made up of 2 hemispheres and a cylinder.

What is the exact volume of this figure?

Answers

Answer:

Hemisphere radius = 3 inch

Sphere volume = 4/3 • π • radius^3

Sphere volume = 4/3 • π • 27

Sphere volume = 113.0973355292 cubic inches

Cylinder Volume = π • radius^2 • height

Cylinder Volume = π • 9 * 10

Cylinder Volume = 282.7433388231 cubic inches

TOTAL VOLUME = 113.10 + 282.74 cubic inches

TOTAL VOLUME = 395.84 cubic inches

Step-by-step explanation:

Answer: 126π in.³

Step-by-step explanation:

The coordinates A(0,0), B(0,3), C(4,3) and D(4,0) are graphed and connected together. What 3 dimensional shape will be created when this shape is rotated about the y-axis?

Answers

a cube would be formed as ABCD form a square when graphed

Which are the maximum values ? I will add a picture please help

Answers

Answer:

58 at the point (9,8)

7 at the point (1, 1)

Step-by-step explanation:

The maximum points will be found in the vertices of the region.

Therefore the first step to solve the problem is to identify through the graph, the vertices of the figure.

The vertices found are:

(1, 10)

(1, 1)

(9, 5)

(9, 8)

We look for the values of x and y belonging to the region, which maximize the objective function [tex]f(x, y) = 2x + 5y[/tex]. Therefore we look for the vertices with the values of x and y higher.

(1, 10), (9, 5), (9, 8)

Now we substitute these points in the objective function and select the one that produces the highest value for f (x, y)

[tex]f(1, 10) = 2(1) + 5(10) = 52\\\\f(9, 5) = 2(9) + 5(5) = 43\\\\f(9, 8) = 2(9) + 5(8) = 58[/tex]

The point that maximizes the function is:

[tex](9, 8)[/tex] with [tex]f(9, 8) = 58[/tex]

Then the value that produces the minimum of f(x, y) is (1, 1)

[tex]f(1, 1) = 2(1) + 5(1) = 7[/tex]

Grace and her brother Sid want to raise money to go to band camp. Their parents have agreed to help them earn up to $400 by paying them $25 when one of them mows the lawn and $10 for each hour that one of them babysits their younger brother. They will have to do a combination of both chores in order to earn the money. Select the equation that represents the number of lawns they can mow, m, and hours they can babysit, b, to earn $400.

Answers

Answer:

25m + 10b = 400

Step-by-step explanation:

Total amount that can be earned = $ 400

Amount earned on mowing a lawn = $ 25

Amount earned per hour on babysitting = $10

Amount earned for mowing one lawn is $ 25, so amount earned for mowing m lawns will be $ 25m

Amount earned for one hour of babysitting is $ 10, so amount earned for babysitting for b hours will be $ 10b

Total amount earned = Amount earned from mowing m lawns + Amount earned from babysitting for b hours

Using the values from above, we can set up the equation as:

400 = 25m + 10b

or

25m + 10b = 400

This equation represents the number of lawns they can mow, m, and hours they can babysit, b, to earn $400.

Leo wants to paint a mural that covers a wall of an area of 600 square feet. The height of the wall is 2/3 of its lengtht. What is the lenght and the height of the wall?

Answers

Length is 30 feet and height is 20 feet.

To determine the dimensions of the wall, we set up an equation using the given area and the fact that the wall's height is 2/3 of its length. Solving for length and height, we find that the length is 30 feet and the height is 20 feet.

The question asks to determine the length and height of a wall that Leo wants to paint, given that the area of the wall is 600 square feet, and the height is 2/3 of its length. To solve this, we will set up an equation to represent the relationship between the length and height and use the area to find the specific dimensions.

Let's use L to represent the length of the wall and H to represent the height. According to the given information, H = (2/3)L. The area of a rectangle is calculated by multiplying its length by its height, so we can set up the equation:

Area = L \H = 600 sq ft

Substituting the height with its relation to length, we get:

600 = L \(2/3)L

Dividing both sides by L and then by 2/3, we find that:

L² = 600 \(3/2)

L² = 900

L = 30 ft (since length must be positive)

Now, to find the height:

H = (2/3)\L = (2/3) \ 30 = 20 ft

So, the length of the wall is 30 feet and the height is 20 feet.

A line passes through the points (1,1/2) and (3,2) which is the equation of the line

Answers

Answer:

The equation of the line is y = 3/4x - 1/4

Step-by-step explanation:

To find the equation of this line, start by using the two points with the slope formula to find the slope.

m(slope) = (y2 - y1)/(x2 - x1)

m = (2 - 1/2)/(3 - 1)

m = (3/2)/2

m = 3/4

Now that we have the slope, we can use that and either point in point-slope form to find the equation.

y - y1 = m(x - x1)

y - 2 = 3/4(x - 3)

y - 2 = 3/4x - 9/4

y = 3/4x - 1/4

Jacob made a banner for a sporting event in the shape of a parallelogram. The area of the banner is 127 1/2 square centimeters. The height of the banner is 4 1/4 centimeters. What is the base of the banner?

Answers

Answer:30

Step-by-step explanation: its 30 because you divide 127 1/2 or 127.5 by 4.25 or 4 1/4.

Assume f(x) = -2x + 8 and g(x) = 3x. What is the value of (g o f)(3)

Answers

Answer:

6

Step-by-step explanation:

(g o f)(3) = g(f(3))

We can find f(3) using its formula with x=3:

f(3) = -2·3 +8 = -6 +8 = 2

Now, we can find g(2) using its formula with x=2:

g(2) = 3·2 = 6

So, the composition value is ...

(g o f)(3) = 6

5x-3y=18
3x+3y=30
Please help

Answers

Solve for x in 5x - 3y = 18

x = 3(6 + y)/5

Substitute x = 3(6 + y)/5 into 3x + 3y = 30

9(6 + y)/5 + 3y = 30

Solve for y in 9(6 + y)/5 + 3y = 30

y = 4

Substitute y = 4 into x = 3(6 + y)/5

x = 6

Therefore,

x = 6

y = 4

The solution to the system of equations is x = 6 and y = 4.

Step 1: Add the two equations.

5x - 3y = 18

3x + 3y = 30

8x = 48

Step 2: Solve for x.

8x = 48

x = 6

Step 3: Substitute x = 6 into one of the original equations to solve for y.

5(6) - 3y = 18

30 - 3y = 18

-3y = -12

y = 4

How would you find the volume of a rectangular prism with length of 1/4 inches, a width of 1 1/4 inches, and height of 4

Answers

Answer:

The volume would be 1 1/4 cubic inches.

Step-by-step explanation:

To find this, multiply all three numbers together because the formula for volume is length multiplied by width multiplied by height.

V = lwh

V = (1/4)(1 1/4)(4)

V = 1 1/4

One hundred people were surveyed about their favorite kind of yogurt. How many more people prefer chocolate than vanilla?
(vanilla- 25%)
(swirl-35%)
(chocolate- 40%)
a. 10
b. 15
c. 25
d. 40

Answers

Answer:

b) 15

Step-by-step explanation:

Given that,

Total people = 100

% of people prefer vanilla = 25%

Number of people prefer vanilla out of 100 = 25% of 100

=> 25/100 * 100 = 25

% of people prefer swirl = 35%

Number of people prefer swirl out of 100 = 35% of 100

=> 35/100 * 100 = 35

% of people prefer chocolate = 40%

Number of people prefer chocolate out of 100 = 40% of 100

=> 40/100 * 100 = 40

The number of more people preferring chocolate over vanilla =  40 - 25

=> 15 so option b is correct

Answer:

The answer is B (15)

Step-by-step explanation:

Pascal wanted the area of the floor to be 54 square feet and the width still be 2/3 the length.What would the dimensions of the floor be?

PLZ HELP ME QUICKLY​

Answers

The dimensions should be 6ft by 9ft.

Final answer:

The dimensions of the floor with an area of 54 square feet, where the width is 2/3 of the length, are 9 feet in length and 6 feet in width.

Explanation:

Pascal wants to find the dimensions of a floor where the area is 54 square feet and the width is 2/3 the length. To find the dimensions, we set up an equation where the length is L and the width is W, with W = (2/3)L. Since area is calculated by the formula Area = length times width, we get 54 = L times (2/3)L. Solving this equation by multiplying the length terms and then dividing both sides by 2/3, we get L² = 54 times (3/2), which simplifies to L² = 81. Taking the square root of both sides gives us L = 9 feet. Consequently, the width would be W = (2/3) times 9 = 6 feet. Therefore, the dimensions of the floor are 9 feet by 6 feet.

PLEASE HELP I'M VERY CONFUSED!
find all natural number solutions for each congruence.

1. y + 3 ≡ 2 mod 4

2. 3 - y ≡ 5 mod 9

Answers

Answer:

1. y = 4n -1

2. y = 9n -2

Step-by-step explanation:

1. For some integer n, numbers that are congruent to 2 mod 4 are ... 4n+2. Then you have ...

y + 3 = 4n +2 . . . . next, subtract 3 from both sides

y = 4n -1 . . . . values of y are natural numbers for all natural numbers n

___

2. Similarly, ...

3 - y = 9n +5

3 - 9n - 5 = y . . . . . add y-9n-5

-9n -2 = y . . . . . . . . collect terms

For every integer n, there is also an integer -n. For our purpose, we're only interested in those values of n that make -9n be positive. Rather than require n be negative so -9n is positive, we can require n be positive and use the expression 9n for the multiple of 9.

We want y a natural number, so we can write this as ...

y = 9n -2 . . . . for natural numbers n

Final answer:

To find the natural number solutions for congruences, we use algebraic manipulation to simplify the equation and find values that satisfy the congruence. For the given congruences, the natural number solutions are y = 3, 7, 11, 15, ... and y = 7, 16, 25, ..., respectively.

Explanation:

1. To find the natural number solutions for the congruence y + 3 ≡ 2 mod 4, we need to find all values of y that satisfy the given congruence.

Subtracting 3 from both sides of the congruence, we have y ≡ -1 mod 4. Since we are looking for natural number solutions, we can rewrite this as y ≡ 3 mod 4.

The natural number solutions for this congruence are y = 3, 7, 11, 15, ...

2. For the congruence 3 - y ≡ 5 mod 9, we need to determine the natural number solutions for y that make the congruence true.

Subtracting 3 from both sides, we have -y ≡ 2 mod 9. Multiplying both sides by -1, we get y ≡ -2 mod 9. Rewriting this as y ≡ 7 mod 9, the natural number solutions for y are y = 7, 16, 25, ...

Learn more about Finding natural number solutions for congruences here:

https://brainly.com/question/12417750

#SPJ3

Complete the square to determine the minimum or maximum value of the function defined by the expression.

x2 − 12x − 2

A) maximum value at 38 B) minimum value at 38 C) maximum value at −38 D) minimum value at −38

Answers

Answer:

Option D. minimum value at −38

Step-by-step explanation:

we have

[tex]x^{2}-12x-2[/tex]

Let

[tex]y=x^{2}-12x-2[/tex]

Complete the square

[tex]y+2=x^{2}-12x[/tex]

[tex]y+2+36=(x^{2}-12x+36)[/tex]

[tex]y+38=(x^{2}-12x+36)[/tex]

[tex]y+38=(x-6)^{2}[/tex]

[tex]y=(x-6)^{2}-38[/tex] ------> equation of a vertical parabola in vertex form

The vertex is the point [tex](6,-38)[/tex]

The parabola open upward-----> the vertex is a minimum

therefore

minimum value at −38

help please

problem 8 and fix problem 10​

Answers

Answer:

8. A'(-1/2, 1), B'(0, 0), C'(1, 1 1/2)

10. 20°, 60°, 100°

Step-by-step explanation:

8. Point B is at the origin, the center of dilation, so its coordinates remain unchanged. The coordinates of A and C are each multiplied by the scale factor, 1/2.

A' = (1/2)A = (1/2)(-1, 2) = (-1/2, 1)

C' = (1/2)C = (1/2)(2, 3) = (1, 3/2)

__

10. The total number of ratio units is 1+3+5 = 9. The total of angle measures in any triangle is 180°. So, each of the "ratio units" must stand for 180°/9 = 20°. Then the angles are ...

1 · 20° = 20°

3 · 20° = 60°

5 · 20° = 100°

Find the value of x in the triangle above Thank you to whoever helps

Answers

this is an isosceles triangle two sides are the same

so there are 180° in a triangle

180-58= 122 now divide this by two because the last two angles are equal

122/2=61° =x°

simplify 6/4-√5
___+___

Answers

Answer:

3 - 2√5

-------------  

      2

Step-by-step explanation:

These two expressions have different denominators, and thus use of the LCD is necessary.  The LCD is 4.

Thus, we have:

6/4 - (4/4)√5, or

6 - 4√5

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

      4

This can be reduced to:

3 - 2√5

-------------   (answer)

      2

The answer to this equation

Answers

Answer:

a = 12

Step-by-step explanation:

Multiply the equation by the inverse of the coefficient of "a".

4·(1/4)a = 4·3

1·a = 12 . . . . . . simplify

a = 12

I need help with a calculus 2 exercise, with a good explanation to what I'm trying to figure out.

The problem involves finding the arc length when y = (x^2)/2 - (lnx/4) given 2≤ x ≤ 4

So I know the formula involves L = Integral from a to b of sqrt(1+(f'(x))^2)dx

And I took the the derivative of y = (x^2)/2 - (lnx/4)
And got = x - (1/4x)
I then kept reducing it to suit the formula
1 + ( x - 1/4x)^2
I added the alike terms
1 + x^2 - 1/2 + 1/16x^2
x^2 + 1/2 + 1/16x^2
AND then comes my question!
So it is supposed to become this afterwards: (x + 1/4x) ^ 2
But how did that happen, I don't understand how I can reduce it to a square, please help me figure it out.

Answers

Explanation:

When writing rational expressions, you need to be aware that ...

  1/4x = (1/4)x ≠ 1/(4x)

Parentheses around the denominator are required, unless you're typesetting the expression and can use a fraction bar for grouping.

The derivative of the curve expression is ...

  y' = x - 1/(4x) . . . . . parentheses added to what you wrote

and the expression (1 -(y')^2) can be written ...

  1 -(y')^2 = x^2 +1/2 +1/(16x^2) . . . . . parentheses added to what you wrote

The first and last terms of this trinomial are both perfect squares, so you might suspect the whole trinomial is a perfect square. You recall that ...

  (a +b)^2 = a^2 + 2ab + b^2

This is a good "pattern" to remember. Using it is a matter of pattern recognition, as is the case with a lot of math.

Here, you have ...

  a = x

  b = 1/(4x)

In order for your trinomial to be a perfect square, the product 2ab must equal the middle term of your trinomial. (Spoiler: it does.)

  2ab = 2(x)(1/(4x)) = (2x)/(4x) = 1/2 . . . . . matches the middle term of 1 -(y')^2

Hence your trinomial can be written as the square ...

  1 -(y')^2 = (x +1/(4x))^2

_____

This is convenient because you want to integrate the square root of this. Your integral then becomes ...

[tex]\displaystyle\int\limits_{2}^{4}{\left(x+\frac{1}{4x}\right)\,dx[/tex]

Find the volume of a cube whose sides are each 6 centimeters long.

A. 216 cm3
B. 360 cm3
C. 644 cm3
D. 1,028 cm3

Answers

Answer:

  A.  216 cm³

Step-by-step explanation:

The volume of a cube is the cube of the side length:

  V = s³ = (6 cm)³ = 216 cm³

Betsy works as waitress. Today she worked an 8 hour shift, and was paid $ 92.00. Plot a graph of the amount Betsy earns against the time she works for. Then find how much Betsy is paid per hour. Write an equation using Y = MX + B Form.

Answers

Final answer:

To plot a graph of the amount Betsy earns against the time she works, we'll put the hours worked on the x-axis and the amount earned on the y-axis. Betsy is paid $11.50 per hour and the equation for Betsy's earnings is Y = 11.50X.

Explanation:

To plot a graph of the amount Betsy earns against the time she works, we'll put the hours worked on the x-axis and the amount earned on the y-axis. In this case, Betsy worked an 8 hour shift and was paid $92.00. So, the point on the graph would be (8, 92). To find how much Betsy is paid per hour, we can divide the total amount earned by the number of hours worked. In this case, $92.00 divided by 8 hours equals $11.50 per hour.

Using the equation Y = MX + B, where Y is the amount earned, M is the rate per hour, X is the number of hours worked, and B is the y-intercept, we can write the equation for Betsy as Y = 11.50X + 0, since she doesn't have a one-time fee.

Other Questions
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