What's the probability of rolling a prime number on a fair six-sided die?

The inverse of f(x) is f⁻¹(x) = (x - 3)/3.

To find the inverse of the function f(x), we need to switch the roles of x and y and solve for y.

Given:

f(x) = 3x + 3

Step 1: Replace f(x) with y:

y = 3x + 3

Step 2: Swap x and y:

x = 3y + 3

Step 3: Solve for y:

x - 3 = 3y

3y = x - 3

y = (x - 3)/3

Therefore, the inverse of f(x) is f⁻¹(x) = (x - 3)/3.

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Answer:

This quadratic function has no roots.

There are no roots of the quadratic function shown in the graph

We we consider the quadratic equation given be of the form ax^2 + bx + c = 0, then

The quantity b^2 - 4ac is known its discriminant.

The solution contains the term [tex]\sqrt{b^2 - 4ac}[/tex] which will be:

Real and distinct if the discriminant is positive

Real and equal if the discriminant is 0

Non-real and distinct roots if the discriminant is negative

Therefore, it has to be two roots of a quadratic equation always(assuming existence of complex numbers). We say that the considered quadratic equation has 2 solutions if roots are distinct, and have 1 solution when both roots are the same.

Hence, we can conclude that there are no roots. so option D is correct.

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The probability that the teacher chooses a girl with glasses is 3/16, and the probability that the teacher chooses a boy with glasses is 1/4.

Let's calculate the probability that the teacher chooses a student with glasses.
a) A girl with glasses: There are 6 girls in the class, and 3 of them wear glasses. There are a total of 16 students (6 girls + 10 boys).
Probability = (Number of girls with glasses) / (Total number of students)
Probability = 3 girls with glasses / 16 total students
Probability = 3/16
b) A boy with glasses: There are 10 boys in the class, and 4 of them wear glasses.
Probability = (Number of boys with glasses) / (Total number of students)
Probability = 4 boys with glasses / 16 total students
Probability = 1/4
So, the probability that the teacher chooses a girl with glasses is 3/16, and the probability that the teacher chooses a boy with glasses is 1/4.

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The median number of pages of the 9 books that Mary read during the summer reading program is  195 pages .

To find the median number of pages of the 9 books that Mary read during the summer reading program, we first need to arrange the pages in ascending order:

112 pages, 138 pages, 138 pages, 159 pages, 195 pages, 210 pages, 217 pages, 270 pages, 356 pages

The median is the middle value in this ordered list. Since there are an odd number of values, the middle value is the fifth one, which is 195 pages.

The median is a measure of central tendency that divides a dataset into two equal halves. In this case, it represents the number of pages that is halfway between the smallest and largest values. It is a useful measure because it is less affected by outliers or extreme values than the mean.

In Mary's case, the median number of pages of the 9 books she read during the summer reading program is 195 pages. This means that half of the books she read had fewer than 195 pages and half had more than 195 pages. It gives us a good idea of what to expect in terms of the length of the books she read.

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The function that corresponds to the parabola Graph is f(x) = -3(x + 1)² + 2, which is option B.

The graph represents a downward open parabola that rises from (-2.4,-4) to (-1,2) and then declines through (0.4,-4) on the x-y coordinate plane.

To determine the function that corresponds to this graph, we need to consider the key features of a parabolic function, including its vertex and axis of symmetry.

The vertex of a parabola in the form f(x) = a(x-h)² + k is (h,k), and the axis of symmetry is x = h

From the graph, we can see that the vertex of the parabola is at (-1,2), which means that h = -1 and k = 2.

We also know that the parabola opens downward, which means that a < 0.

Therefore, we can eliminate options A and D since they both have a positive value of "a."

Next, we can test options B and C.

Option B has the form f(x) = -3(x + 1)² + 2, which means that the vertex is at (-1,2) and the parabola opens downward due to the negative coefficient of (x+1)².

To confirm if option B is correct, we can check if the point (0.4,-4) lies on the parabola:

f(0.4) = -3(0.4+1)² + 2 = -3(1.4)² + 2 = -5.88

Since the y-coordinate of the point (0.4,-4) is -4, which is equal to -5.88, we can see that the point lies on the parabola.

Therefore, the function that corresponds to the graph is f(x) = -3(x + 1)² + 2, which is option B.

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Answer:

midsegment = 20

Step-by-step explanation:

the midsegment is half the sum of the bases, that is

midsegment = [tex]\frac{12+28}{2}[/tex] = [tex]\frac{40}{2}[/tex] = 20

The determinant of the given matrix is -43.

We are given that;

The matrix= 3 2 -2

1 2 4

-1 3 2

Now,

To evaluate the determinant of the matrix

[3  2 -2]

[1  2  4]

[-1 3  2]

you can use the rule of Sarrus. First, you need to copy the first two columns of the matrix and place them next to the third column:

[3  2 -2 | 3  2]

[1  2  4 | 1  2]

[-1 3  2 | -1 3]

Then, you can calculate the sum of the products of the three diagonals going from left to right:

(3 * 2 * 2) + (2 * 4 * -1) + (-2 * 1 * 3) = 12 -8 -6 = -2

and subtract from it the sum of the products of the three diagonals going from right to left:

(-1 * 2 * -1) + (3 * 4 * 3) + (2 * 1 * 2) = 1 +36 +4 =41

-2 -41 = -43.

Therefore, by the matrices the answer will be -43.

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I recommend use of simple random sampling to determine the three possible toppings to be used for the vegetarian pizza.

It is a sampling technique in which each member of the population has an equal chance of being selected for the sample. Here, the population would be students who regularly eat at the school cafeteria.

By using the sampling, we make sure that each student has an equal chance of being selected to taste test the different pizza toppings. This will help to eliminate bias and provide a representative sample of the population's preferences. They will provide feedback which will be used to determine which topping should be used for the vegetarian pizza.

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The Earth's human population, approximately 7,900,000,000 people, can be expressed using scientific notation as 7.9 x 10^9.

Scientific notation is used to express extremely large numbers using a number between 1 and 10.

In this case, that number is 7.9

We need to move that decimal spot 9 places to the right to get 7,900,000,000. Therfore, to state that number in scientific notation, we would write it as:

7.9 * 10^9

~~~Harsha~~~

If Angle 2 Is-congruent-to Angle 6, then the statement which must be true about Angle 2 is that Angle 2 is supplementary to Angle 8.

We, have,

This refers to the type of angle where two different angles are said to have equal measure on their corresponding sides.

With this in mind, we can see that based on the two horizontal parallel lines are intersected by a third line, there is a third line that intersects the top line and forms four angles.

Hence, if If Angle 2 Is-congruent-to Angle 6, then the statement which must be true about Angle 2 is that Angle 2 is supplementary to Angle 8 because the sum of either angles is 180 degrees.

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complete queston:

Two lines are intersected by a third line.

Two horizontal parallel lines are intersected by a third line. The third line intersects the top line and forms 4 angles. Labeled clockwise, from uppercase left, the angles are 1, 2, 4, 3. The third line intersects the bottom line and forms 4 angles. Labeled clockwise, from uppercase left, the angles are 5, 6, 8, 7.

If Angle2 Is-congruent-to Angle6, which must be true about Angle2?

Angle2 Is-congruent-to Angle5

Angle2 is complementary to Angle5.

mAngle2 = mAngle8

Angle2 is supplementary to Angle8.

Answer: The standard equation of a parabola is given by:

4p(x - h) = (y - k)^2

where (h, k) is the vertex of the parabola, p is the distance from the vertex to the focus and also the distance from the vertex to the directrix.

In this case, the focus is at (-2,1), so the vertex is halfway between the focus and the directrix, which is at (x, y) = (-2, (1+8)/2) = (-2, 4.5). The distance from the vertex to the focus (and to the directrix) is p = 3.5.

Substituting these values into the equation, we get:

4(3.5)(x + 2) = (y - 4.5)^2

Simplifying, we get:

14(x + 2) = (y - 4.5)^2

This is the standard equation of the parabola.

To sketch the graph, we can use the vertex (-2, 4.5) as a starting point, and use the distance from the vertex to the focus to plot some points on either side of the vertex. Since the focus is to the left of the vertex, we know that the parabola will open to the left.

Using the distance formula, we can find the coordinates of two points on the parabola that are equidistant from the focus and the directrix:

Point 1: (-5.5, 4.5)

Distance from focus: sqrt((-5.5 + 2)^2 + (4.5 - 1)^2) = sqrt(44.5) ≈ 6.67

Distance from directrix: |4.5 - 8| = 3.5

Point 2: (-5.5, 1.5)

Distance from focus: sqrt((-5.5 + 2)^2 + (1.5 - 1)^2) = sqrt(36.5) ≈ 6.04

Distance from directrix: |1.5 - 8| = 6.5

Parabola with a focus at (-2,1) and directrix y=8

The vertex is labeled as V, the focus as F, and the directrix as D. The distance from the vertex to the focus (and directrix) is labeled as p. The parabola opens to the left and is symmetric about the axis of symmetry, which is the vertical line passing through the vertex.

y=3(5)ˣ is the equation of the function represented by the table

We can see that as x increases, y increases exponentially.

To determine the specific form of the exponential function, we can take the ratio of successive y-values:

(3/5)/(3/25) = 5/1

(3)/(3/5) = 5

(15)/(3) = 5

(75)/(15) = 5

This shows that the ratio of successive y-values is constant at 5. Therefore, the function represented by the table is an exponential function of the form:

y = a(b)ˣ

To find a and b, we can use any two pairs of (x,y) values.

Let's use (0,3) and (1,15):

3 = a(b)⁰

15 = a(b)¹

3 = a

Now we get b =5

Hence, y=3(5)ˣ is the equation of the function represented by the table

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The pH level of the pool is determined as 8.6.

The pH of a solution is a measure of its acidity or basicity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral, less than 7 is acidic and above 7 is alkaline.

The pH of a solution can be calculated using the formula;

pH = -log[H₃O⁺]

where;

Using the measured [H₃O⁺] concentration of 2.40 x 10⁻⁹ moles/liter, we can calculate the pH of the swimming pool as follows;

pH = -log(2.40 x 10⁻⁹)

pH = 8.62

Therefore, the pH of the swimming pool is 8.6, rounded to the nearest tenth.

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Hello !

Answer:

198 apples

Step-by-step explanation:

12 apples <=> 7 pounds of dough

x apples <=> 115 pounds of dough

Let's use a cross product :

[tex]x = \frac{12 \times 115}{7} \approx197.14 [/tex]

198 apples are needed.

Have a nice day

Answer:

Not rational

Step-by-step explanation:

A rational number is any integer, fraction, terminating decimal, or repeating decimal.

Anything with a square root is automatically not a rational number.

have a great day and thx for your inquiry :)

Total length of the two sides of the triangle is 2x² + 4x + 4 and  length of the third side of the triangle is 5x³ - 5x² + 3x - 12.

The total length of the two sides of the triangle is the sum of Side 1 and Side 2:

(3x² - 4x - 1) + (4x - x² + 5)

2x² + 4x + 4

The total length of the two sides of the triangle is 2x² + 4x + 4.

The length of the third side of the triangle can be found by subtracting the sum of Side 1 and Side 2 from the perimeter of the triangle:

Perimeter - (Side 1 + Side 2)

(5x³ - 2x² + 3x - 8) - (3x² - 4x - 1 + 4x - x² + 5)

Combining like terms

5x³ - 5x² + 3x - 12

The length of the third side of the triangle is 5x³ - 5x² + 3x - 12.

The polynomials are closed under addition and subtraction by part A and part B because when we added Side 1 and Side 2, and when we subtracted their sum from the perimeter of the triangle, the resulting expressions were still polynomials with real coefficients.

Therefore, the sum and difference of polynomials with real coefficients are also polynomials with real coefficients.

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The solution to the set {2x < 6} n {x - 5 > -4} is {x | 1 < x < 3 }

From the question, we have the following parameters that can be used in our computation:

{2x < 6} n {x - 5 > -4}

Divide both sides of 2x < 6 by 2

So, we have

{x < 3} n {x - 5 > -4}

Add 5 to both sides of x - 5 > -4

So, we have

{x < 3} n {x > 1}

The above set when combined is

1 < x < 3

This means that the solution to the set is {x | 1 < x < 3 }

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The sum of the equation 3y/y²+7y+10 + 2/y+2 is 5/(y + 5).

In order to find the sum, we have to first find a common denominator for the two fractions:

3y/y² + 7y + 10 + 2/y + 2 = (3y/(y + 5)(y + 2)) + (2(y + 5)/(y + 5)(y + 2))

Next, we join the two fractions by adding their numerators:

= (3y + 2(y + 5))/((y + 5)(y + 2))

Then, we simplify the numerator:

= (3y + 2y + 10)/((y + 5)(y + 2))

= (5y + 10)/((y + 5)(y + 2))

= 5(y + 2)/((y + 5)(y + 2))

= 5/(y + 5)

Therefore, the sum of 3y/y²+7y+10 + 2/y+2 = 5/(y + 5).

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patterns a and d have a constant difference, pattern c has a constant difference, and patterns b, e, and f do not have a constant difference or a constant ratio.

a. The pattern has a constant difference. The first term is 1, the second term is 4, which is 3 more than 1, the third term is 10, which is 6 more than 4, and the fourth term is 19, which is 9 more than 10.

b. The pattern does not have a constant difference or a constant ratio. The differences between consecutive terms are 3, 2, and 4, respectively, which are not constant. The ratios between consecutive terms are 2.5/2, 1.4/2.5, and 11/1.4, respectively, which are also not constant.

c. The pattern has a constant difference. The first term is 3, the second term is 7, which is 4 more than 3, the third term is 13, which is 6 more than 7, and the fourth term is 21, which is 8 more than 13.

d. The pattern has a constant difference. The first term is 12, the second term is 10, which is 2 less than 12, the third term is 6, which is 4 less than 10, and the fourth term is 0, which is 6 less than 6.

e. The pattern does not have a constant difference or a constant ratio. The differences between consecutive terms are 4, 7, and 10, respectively, which are not constant. The ratios between consecutive terms are 2.5/2, 13/2.5, and 23/13, respectively, which are also not constant.

f. The pattern does not have a constant difference or a constant ratio. The differences between consecutive terms are 6, 12, and 18, respectively, which are not constant.

The ratios between consecutive terms are 13/7, 25/13, and 43/25, respectively, which are also not constant.

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Using a trigonometric relation we can see that the distance between the base and the wall is 26.76ft

In the image at the end you can see a diagram for this problem.

We have a right triangle where we want to find the value of d, which is the adjacent cathetus to the known angle.

Then we can use the trigonometric relation:

cos(a) = (adjacent cathetus)/hypotenuse

Where:

a = 48°

hypotenuse = 40ft

adjacent cathetus = d

Replacing that we will get:

cos(48°) = d/40ft

Solving that for d, we will get:

40ft*cos(48°) =d

26.76ft = d

That is the distance between the base and the wall.

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In the given diagram, the area of the shape is 2π units²

From the question, we are to calculate the area of the given shape.

In the given diagram, the shape is a quadrant.

The area of a quadrant is given by the formula

Area = 1/4 πr²

Where r is the radius

From the given information,

Radius = 8

Thus,

Area of the quadrant = 1/4 × π × 8

Area of the quadrant = 2π units²

OR

Area of the quadrant = 2 × 3.14 units²

Area of the quadrant = 6.28 units²

Hence,

The area of the shape is 2π units² OR 6.28 units².

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The outcomes in set A U B are 1, 3, 5, 6, 7, 8, 9, 10, 11, 12 (the odd sums) 4, 8, 12 (the sums divisible by 4)

The sum of the number cubes is odd if one die shows an even number and the other shows an odd number, or vice versa.

There are 3 even numbers and 3 odd numbers on a number cube, so there are 3 x 3 = 9 ways to get an odd sum.

The sum of the number cubes is divisible by 4 if both dice show either 2 or 4.

There are 2 ways to get a 2 on a number cube and 2 ways to get a 4, so there are 2 x 2 = 4 ways to get a sum of 4.

Therefore, the outcomes in A U B are 1, 3, 5, 6, 7, 8, 9, 10, 11, 12 (the odd sums) 4, 8, 12 (the sums divisible by 4)

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To have an infinite number of solutions the value of a would be

D. 20

Information given in the problem includes

5(x - 3) + a = 5(x + 1)

To have an infinite number of solutions we can simplify as follows

5(x - 3) + a = 5(x + 1)

opening the parenthesis

5x - 15 + a = 5x + 5

collecting like terms

-15 + a = 5

Adding 15 to both sides, we get:

a = 20

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Answer:

GJ ≈ 31.42 units

Step-by-step explanation:

the arc length GJ is calculated as

GJ = circumference of circle × fraction of circle

     = 2πr × [tex]\frac{30}{180}[/tex] ( r is the radius )

the radius HJ = 30 , then

GJ = 2π × 30 × [tex]\frac{1}{6}[/tex]

     = 60π × [tex]\frac{1}{6}[/tex]

     = 10π

     ≈ 31.42 units ( to the nearest hundredth )

Using a pattern we can find the missing values in the table, these are:

When you look at the table, you can see that for each increase of 1 unit in the value of x, we have an increase of 14 units in the value of y. So we have a really simple pattern:

f(x) = f(x - 1) + 14

Then to get the next values in the table, just add 14 to the previous ones.

A = 42 + 14 = 56

B = 56 + 14 = 70

C = 70 + 14 = 84

Thes are the 3 missing values in the table.

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The solution to the system of equations, using the Gauss-Jordan method, is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&14.28\\0&1&0&-21.4\\0&0&1&1.2\end{array}\right][/tex]

The matrix representing the system of equations is given as follows:

[tex]\left[\begin{array}{cccc}-5&-3&-4&-12\\0&-2&-7&38\\0&1&4&-22\end{array}\right][/tex]

First we want a value of 1 at line 1, column 1, hence we multiply the first line by -1/5, that is:

R1 -> -1/5R1

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&-2&-7&38\\0&1&4&-22\end{array}\right][/tex]

First we want a value of 1 at line 2, column 2, hence we multiply the second line by -1/2, that is:

R2 -> -1/2R2

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&1&1.5&-19\\0&1&4&-22\end{array}\right][/tex]

We want an element of zero at line 3, column 2, hence:

R3 -> R3 - R2

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&1&1.5&-19\\0&0&2.5&-3\end{array}\right][/tex]

First we want a value of 1 at line 3, column 3, hence we multiply the third line by 2.5, that is:

R3 -> 1/2.5R3

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&1&1.5&-19\\0&0&1&1.2\end{array}\right][/tex]

Then the solution to the system of equations is given as follows:

Hence the row-echelon form is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&14.28\\0&1&0&-21.4\\0&0&1&1.2\end{array}\right][/tex]

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Since 6 is less than 84, the statement is correct. The roof with dimensions 12 meters by 7 meters is sufficient to accommodate 6 solar panels, each with a width of 1 meter.

To verify Sam's statement, we can calculate whether a roof that is 12 meters long and 7 meters wide can accommodate 6 solar panels, with each panel having a width of 1 meter.

The total area required to install 6 solar panels with a width of 1 meter each is:

6 panels * 1 meter = 6 square meters

Now, let's calculate the area of the roof:

Area of the roof = length * width = 12 meters * 7 meters = 84 square meters

Since the area required for the solar panels is 6 square meters and the area of the roof is 84 square meters, we can determine if the statement is correct.

If 6 square meters is less than or equal to 84 square meters, then the roof is indeed enough to install the 6 solar panels.

6 square meters ≤ 84 square meters.

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The value of missing side length is, 51.11

We have to given that;

A triangle is shown.

And, To find the missing side of triangle.

Since, An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.

And, A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

Now, By using trigonometry formula, we get;

⇒ cos 63° = Hypotenuse / Base

⇒ cos 63° = 23 / x

⇒ 0.45 = 23 / x

⇒ x = 23 / 0.45

⇒ x = 51.11

Therefore, The value of missing side in the triangle is,

⇒ x = 51.11

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Answer:

a) 2/25. b) 3/100. c) 1/20. d) 11/50.

Step-by-step explanation:

there are 3 + 6 + 4 + 12 = 25 markers

a) p(red) = 4/25. then p(blue) = 12/24.

4/25 X 12/24 = 2/25

b) p(yellow) = 6/25. then p(green) = 3/24.

6/25 X 3/24 = 3/100

c) p(yellow) = 6/25. p(2nd yellow) = 5/24

6/25 X 5/24 = 1/20

d) p(blue) = 12/25. p(2nd blue) = 11/24

12/25 X 11/24 = 11/50

Answer:

√3/2

Step-by-step explanation:

Easy Method

The equation above is in the forms of sin(a)cos(b) - cos(a)sin(b), which is sin(a-b) according to the trig identities. sin(75-15) = sin(60) = √3/2

Harder Method

Find sin75 with equation sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

sin75°

= sin(45° + 30°)

= [sin45°cos30° + cos45°sin30°]

= [√2/2 * √3/2 + √2/2 * 1/2]  <-- unit circle known values

= [(√6 + √2)/4]

Find cos75 with the equation: cos(a+b) = sin(a)sin(b) - cos(a)cos(b)

cos75°

= cos(45° + 30°)

= [cos45°cos30° - sin45°sin30°]

= [√2/2 * √3/2 - √2/2 * 1/2]  <-- unit circle known values

= [(√6 - √2)/4]

Find sin15 with the equation: sin(a-b) = sin(a)cos(b) - cos(a)sin(b)

sin15°

= sin(45° - 30°)

= [sin45°cos30° - cos45°sin30°]

= [√2/2 * √3/2 - √2/2 * 1/2]  <-- unit circle known values

= [(√6 - √2)/4]

Find cos15 with the equation: cos(a-b) = sin(a)sin(b) + cos(a)cos(b)

cos15°

= cos(45° - 30°)

= [cos45°cos30° + sin45°sin30°]

= [√2/2 * √3/2 + √2/2 * 1/2]  <-- unit circle known values

= [(√6 + √2)/4]

Now plug in all the solved values, we get: {[(√6 + √2)/4] * [(√6 + √2)/4]} - {[(√6 - √2)/4] * [(√6 - √2)/4]} = √3/2