The 2 people from the 9 people can sit on the aisle in 72 ways
How many ways can the 2 people sit on the aisle?from the question, we have the following parameters that can be used in our computation:
Number of peiple = 9
Number of seats = 2
Using the above as a guide, we have the following:
n = 9 and r = 2
The can arrange themselves in nPr ways.
Where
nPr = n!/(n - r)!
Substitute the known values in the above equation, so, we have the following representation
9P2 = 9!/(9 - 2)!
Evaluate the expression
9P2 = 72
Hence, the number of ways they can arrange themselves is 72
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Write the trigonometric ratio as a simplified fraction.
10. sin B
C9
A
C
6
15 C
B
11. cos A
12. tan A
10.
11.
12.
The value of the trigonometric ratios are;
sin A = 2/5
cos A = 3/5
tan A = 2/3
How to determine the ratiosIt is important to note that there are six different trigonometric identities and their ratios.
We have;
sine cosinetangentcotangentsecantcosecantFrom the diagram shown, we have;
sin θ = opposite/hypotenuse
cos θ = adjacant/hypotenuse
tan θ = opposite/adjacent
Then,
sin A = 6/15 = 2/5
cos A = 9/15 = 3/5
tan A = 6/9 = 2/3
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The value of the trigonometric ratios are;
sin A = 2/5
cos A = 3/5
tan A = 2/3
How to determine the ratiosIt is important to note that there are six different trigonometric identities and their ratios.
We have;
sine cosinetangentcotangentsecantcosecantFrom the diagram shown, we have;
sin θ = opposite/hypotenuse
cos θ = adjacant/hypotenuse
tan θ = opposite/adjacent
Then,
sin A = 6/15 = 2/5
cos A = 9/15 = 3/5
tan A = 6/9 = 2/3
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Johnny’s Donut Shop is running a “buy one get one 50% off” sale today. During the sale, Laine bought 4 glazed donuts for only $3.78. What is the regular price for one glazed donut at Johnny’s Donut Shop?
Answer:
Step-by-step explanation:
Set up a simple system.
Price = 4x where x = price of donut
For the sale, the price = 3.78, but the x value is 0.5x as it is half off.
$3.78 = 4(0.5x)
Solve for x =$1.89
Therefore, the original price of the donuts were $1.89.
Check by doing 4 x 50%(.5) of 1.89 which equals 3.78!
Which two lanes are parallel
1. p || q
2. r || s
3. q || r
4. p || r
Answer:
4
Step-by-step explanation:
1p:80°-140。＝40°，40°
Plot - 1 1/4 and 5/2 on the number line below.
The number line where we plotted 1 1/4 and 5/2 is added as an attachment
Plotting -1 1/4 and 5/2 on a number lineFrom the question, we have the following parameters that can be used in our computation:
-1 1/4 and 5/2
To start with, we convert both numbers to the same form
i.e. decimal or fraction
When converted to fractions, we have
-5/4 and 5/2
Rewrite the denominators of the fractions
So, we have
-5/4 and 10/4
This means that we can plot -5/4 at -5 and 10/4 at point 10 where the difference in each interval is 1/4
Using the above as a guide, we have the following:
The number line is attached
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Triangle P is rotated 180° about (0, 0) to give triangle Q.
Triangle Q is translated by 5 -2
to give triangle R.
(a) Describe fully the single transformation that maps triangle P onto triangle R.
Answer:
The single transformation that maps triangle P onto triangle R is a reflection about the origin followed by a translation of 5 units to the right and 2 units down.
Denzel used the spinner shown below to compare theoretical probability and experimental probability. He spun the spinner 180 times and recorded the letter that it landed on each time. The results are shown in the table.
Which statement correctly compares the theoretical probability and experimental probability for one of the letters on the spinner?
A. The theoretical probability of landing on a C is greater than the experimental probability of landing on an C.
B. The theoretical probability of landing on a D is greater than the experimental probability of landing on an D.
C. The theoretical probability of landing on a B is less than the experimental probability of landing on an B.
D. The theoretical probability of landing on an A is less than the experimental probability of landing on an A.
Comparing the two probabilities option d: The theoretical probability of landing on an A is less than the experimental probability of landing on an A. is correct.
What is theoretical probability?The chance of an event based solely on mathematical precepts or hypotheses, without any actual experiments or observations, is known as theoretical probability. Probability theory frequently uses theoretical probability to create predictions about the chance of events occurring based on presumptions or models.
On the other hand, experimental probability is the likelihood of an event based on real observations or experiments. A set of trials or experiments are conducted to determine the experimental probability, and the number of times the event occurs during those trials is counted.
For the given spinner the theoretical probability of landing on any one letter is:
P(one letter) = 1/4
Now, experimental probability is given as:
Experimental probability of A = 36/180 = 0.2
Experimental probability of B = 48/180 = 0.2667
Experimental probability of C = 51/180 = 0.2833
Experimental probability of D = 45/180 = 0.25
Thus, comparing the two probabilities option d: The theoretical probability of landing on an A is less than the experimental probability of landing on an A. is correct.
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Please help me with these 6th grade math questions. Thank you so much!
The number line is shown below:
How to solveThe given inequality:
x -5 < 11
Add 5 on both sides
x -5 + 5 < 11 + 5
Add x < 16
Thus, this is represented in the number line below:
A graphical illustration of numbers in a line is known as a number line. It is utilized for indicating the relation between values, and also to do calculations such as subtraction, multiplication, division, etc. The figures get bigger from left to right and diminishes from right to left.
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Answer the question below
Answer:
B represents a function.
A quadratic function has x-intercepts at -7 and 1. The vertex of the function is at (-3, 16). What is the equation of the function?
A. f(x) = x^2 + 6x -7
B. f(x) = -x^2 - 6x+ 7
C. 7(x)=x^2 - 6x + 7
D. f(x)=-x^2 + 6x -7
[tex]f(x) = x^2 + 6x - 7[/tex] is the equation of the function.
What is the function?A relationship between a group of inputs and one output is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function.
To find the equation of the quadratic function, we need to use the vertex form of the equation:
[tex]f(x) = a(x - h)^2 + k[/tex]
where (h, k) is the vertex of the parabola.
We know that the vertex is (-3, 16), so h = -3 and k = 16. We also know that the x-intercepts are at -7 and 1, which means that the factors of the quadratic function are (x + 7) and (x - 1).
To find the value of 'a', we can use either of the x-intercepts:
[tex]f(-7) = a(-7 + 3)^2 + 16 = 0\\\\16a = 16\\\\a = 1[/tex]
Therefore, the equation of the quadratic function is:
[tex]f(x) = (x + 7)(x - 1)\\\\f(x) = x^2 + 6x - 7[/tex]
So the answer is (A)[tex]f(x) = x^2 + 6x - 7[/tex].
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The stem-and-leaf plot displays the amount of time, in minutes, that 10 players spent practicing their serving skills for an upcoming tennis tournament.
1 2, 2, 9
2 0, 2, 6
3 1, 1
4 0
5
6 8
Key: 3|1
means 31
Part A: Calculate the mean, median, mode, and range for the data given. Show your work. (8 points)
Part B: Should the tennis players report the mean or median value to show they are less prepared for the tournament? Explain based on the scenario. (4 points)
Part A:
To calculate the mean, we add up all the values and divide by the total number of values:
12 + 12 + 19 + 20 + 22 + 26 + 31 + 31 + 40 + 68 = 281
Mean = 281 / 10 = 28.1
To find the median, we need to find the middle value. Since there is an even number of values, we take the average of the two middle values:
Median = (22 + 26) / 2 = 24
The mode is the value that appears most frequently. In this case, there is no mode as no value appears more than once.
The range is the difference between the highest and lowest values:
Range = 68 - 12 = 56
Part B:
The median may be a better measure of central tendency in this scenario as there is an outlier value of 68 that can skew the mean. Since outliers can have a significant impact on the mean, reporting the median may give a better representation of the typical amount of time spent practicing for the tournament. Therefore, the tennis players should report the median value.
Why do polling companies often survey 1060 individuals when they wish to estimate a population proportion with a margin of error 3% with 95% confidence?
Polling companies survey 1060 individuals to estimate a population proportion with a margin of error of 3% with 95% confidence because it strikes a balance between cost and accuracy, and it is based on statistical calculations that aim to achieve a reasonable level of precision while minimizing the margin of error.
When a polling company selects a sample size of 1060 individuals, it is based on statistical calculations that aim to achieve a 3% margin of error with 95% confidence. The sample size needs to be large enough to reduce the margin of error and increase the level of confidence in the results.
The margin of error decreases as the sample size increases, and it is inversely proportional to the square root of the sample size. Therefore, a larger sample size leads to a smaller margin of error. However, increasing the sample size also increases the cost and time required to conduct the survey.
Polling companies aim to strike a balance between the cost and the desired level of accuracy, which is typically a margin of error of 3-5% with 95% confidence. A sample size of 1060 individuals is often considered a reasonable compromise between cost and accuracy, as it provides a small enough margin of error to be useful while being feasible to conduct within a reasonable time and budget.
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Find the equation of the plane passing through the lines of intersection of the planes :
2x - 7y + 4z = 3 , 3x - 5y + 4z + 11 = 0 and the point ( -2 , 1 , 3 ).
The equation of the plane passing through the lines of intersection of the planes 2x - 7y + 4z = 3 and 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3) is
-13x - 8y + z + 90 = 0
We have,
To find the equation of the plane passing through the intersection of two planes, we need to first find the direction vector of the line of intersection of the two planes.
Let's call the two planes P1 and P2, which are given by the equations:
P1: 2x - 7y + 4z = 3
P2: 3x - 5y + 4z + 11 = 0
To find the direction vector of the line of intersection of P1 and P2, we can take the cross product of the normal vectors of P1 and P2. The normal vectors are:
n1 = <2, -7, 4>
n2 = <3, -5, 4>
Taking the cross-product of n1 and n2.
n1 x n2 = <-13, -8, 1>
This vector is parallel to the line of intersection of P1 and P2. We can use this vector as the direction vector for the line.
Now, we need to find a point on the line.
We can use the point of intersection of P1 and P2 as a point on the line.
To find this point, we can solve the system of equations:
2x - 7y + 4z = 3
3x - 5y + 4z + 11 = 0
Subtracting the first equation from the second, we get:
x + 2y + 7 = 0
Solving for x in terms of y,.
x = -2y - 7
Substituting this expression for x into the first equation, we get:
-4y - 14 - 7y + 4z = 3
Simplifying.
-11y + 4z = 17
Choosing y = 1.
-11 + 4z = 17
Solving for z.
z = 7
Substituting y = 1 and z = 7 into the equation for x.
x = -2(1) - 7 = -9
So a point on the line is (-9, 1, 7).
Now, we can use the point (-9, 1, 7) and the direction vector <-13, -8, 1> to write the equation of the plane passing through the line of intersection of P1 and P2.
Let's call this plane P3.
Using the point-normal form of the equation of a plane.
-13(x + 9) - 8(y - 1) + (z - 7) = 0
Expanding and simplifying, we get:
-13x - 8y + z + 90 = 0
Thus,
The equation of the plane passing through the lines of intersection of the planes 2x - 7y + 4z = 3 and 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3) is:
-13x - 8y + z + 90 = 0
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For a certain company, the cost function for producing x items is C(x)=50x+250
and the revenue function for selling x items is R(x)=−0.5(x−120)2+7,200
. The maximum capacity of the company is 170
items.
Profit when producing 70
items=?
Profit when producing 80
items=?
Can you explain, from our model, why the company makes less profit when producing 10 more units?
Answer:
Profit when producing 70 items = $3,050
Profit when producing 80 items = $1,350
Step-by-step explanation:
To find the profit when producing a certain number of items, we need to subtract the cost from the revenue:
Profit = Revenue - Cost
Let's calculate the profit for producing 70 items:
Revenue when producing 70 items:
R(70) = -0.5(70-120)^2 + 7,200 = $6,800
Cost of producing 70 items:
C(70) = 50(70) + 250 = $3,750
Profit when producing 70 items:
Profit = Revenue - Cost = $6,800 - $3,750 = $3,050
Similarly, let's calculate the profit for producing 80 items:
Revenue when producing 80 items:
R(80) = -0.5(80-120)^2 + 7,200 = $5,600
Cost of producing 80 items:
C(80) = 50(80) + 250 = $4,250
Profit when producing 80 items:
Profit = Revenue - Cost = $5,600 - $4,250 = $1,350
We can see that the profit is less when producing 10 more units because the cost of producing those additional units exceeds the revenue generated from selling them. In other words, the marginal cost (the cost of producing one additional unit) is greater than the marginal revenue (the revenue generated from selling one additional unit) beyond a certain point.
This is an example of the law of diminishing returns, which states that as we increase the quantity of inputs while keeping other inputs constant, the marginal product (output per unit of input) eventually decreases.
Hope this helps!
you have $5000 toward the purchase of a small boat that will cost $6000. how long will it take the $5000 to grow to $6000 if it is invested at 5% compounded monthly? solve algebraically. round to the nearest hundredth.
Convert the rectangular coordinates to polar coordinates with
r > 0 and 0 ≤ < 2.
(1, −2)
Help I'm stuck with this one :(
If you want to transform rectangular coordinates (x, y) into its polar counterpart (r, θ), do the following:
The StepsFirstly, obtain r by computing the distance from the origin to point (x,y), making use of the Pythagorean theorem or simply put, by evaluating r = sqrt(x^2 + y^2).
Secondly, acquire θ which denotes the angle formed between the positive x-axis and a line correlating the origin and the coordinate (x,y). Calculate this via inverse tangent function: θ = atan2(y, x).
Thirdly, in case the measurement for θ is indicated in radians while you require degrees instead, multiply by 180/π to convert it.
Usually expressed as (r, θ), where r indicates proximity disentangled between the origin and the point (x,y), whilst θ identifies the aforementioned angle.
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Can y’all please help me with this whole thing :/
Answer:
Slope-Intercept Form: y=2x+1
Standard Form: 2x-y=6
1. [tex]y=\frac{1}{3}x-1[/tex]
2. [tex]y=-\frac{2}{5}x[/tex]
3. [tex]y=x+8[/tex]
4. [tex]y=-\frac{1}{2}x+3[/tex]
5. [tex]y=2x+1[/tex]
6. [tex]y=-4x-5[/tex]
7. [tex]y=-x+2[/tex]
8. [tex]y=\frac{2}{3}x-4[/tex]
9. [tex]y=x\\[/tex]
Pls help me by 11 pm
Answer:
Surface Area = 2×(3.5×10 + 3.5×8 + 10×8) = 286 feet
Step-by-step explanation:
First, multiply 3.5 by 10 (35)
Secondly, multiply 3.5 by 8 (28)
Then, multiply 10 by 8 (80)
Next, add all the values together (143)
And lastly, multiply 143 by 2 (286)
Please turn this into your own work, (By making it look like it's worked out) since they can catch you for Plagiarism.
Based on the graph below, what will be the coordinates of the point that represents the number of snowballs needed to make 2 snowmen?
the coordinates of the point that represents the number of snowballs needed to make 2 snowmen should be option C. which is (2,6)
17. La inscripción en una clase de negocios disminuyó de 25 a 15 estudiantes. ¿Cuál fue el porcentaje de disminución en la inscripción
Answer:
40%
Step-by-step explanation:
Lauren over-filled the homemade pecan pie that she was baking for Thanksgiving, so the pie needed additional cooking time. Lauren decided to place a strip of aluminum foil around the edge of the crust so that it would not burn. If Lauren used a pie pan with a 12-inch diameter, how long, to the nearest inch, should the strip of foil be?
A. 19 inches
B. 24 inchess
C. 113 inchess
D. 38 inchess
Hence the correct option is D. 38inches long, to the nearest inch, should the strip of foil be.
What is the circumference?The circumference of a circle in mathematics is its perimeter. That is, if the circle were opened up and straightened out to a line segment, the circumference would be the length of the arc. In according to broader the perimeter is the length of any closed figure of curve.
What is diameter?A straight line and cuts through the middle of a body or figure. particularly the length of a diameter is a line segment passing through the center of a circle with its ends on the circumference.
We determine the circumference of the pie pan in order to determine the length of the aluminium foil strip required.
Circumference =2[tex]\pi[/tex]r.
where, r = circle of radius and [tex]\pi[/tex] = mathematical constant =3.14.
The pie pan of radius = half of its diameter=12 inches/2= 6 inches.
Now, the pie pan of circumference is:
Circumference: 2 [tex]\pi[/tex]r=2[tex]\pi[/tex] *6 inches= 37.7 inches
Therefore , the length of the aluminium foil strip = 38 inches,
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how do i work out 320.041 - 47.96
To subtract 47.96 from 320.041, you can align the decimal points and then subtract each digit from right to left.
First, write the numbers with the decimal points aligned:
320.041
- 47.960
Next, subtract the digits in the ones place, which is 1 minus 0, resulting in 1. Then, subtract the digits in the tenths place, which is 4 minus 6. Since 4 is less than 6, you need to borrow 1 from the digit to the left, making the 2 into a 1 and adding 10 to the 4, giving you 14. So, 14 minus 6 equals 8.
Continue subtracting each digit in the same way:
320.041
- 47.960
= 272.081
Therefore, 320.041 minus 47.96 is equal to 272.081.
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Triangle NMO has vertices at N(−5, 2), M(−2, 1), O(−3 , 3). Determine the vertices of image N′M′O′ if the preimage is reflected over x = −2. N′(1, 2), M′(−2, 1), O′(−1, 3) N′(−5, −6), M′(−2, −5), O′(−3, −7) N′(−5, 0), M′(−2, −1), O′(−3, 1) N′(9, 2), M′(6, 1), O′(7, 3)
The vertices of image N′M′O′ if the preimage is reflected over x = −2 include the following: D. N′(9, 2), M′(6, 1), O′(7, 3)
What is a reflection over the y-axis?In Mathematics and Geometry, a reflection over or across the y-axis or line x = 0 is represented and modeled by this transformation rule (x, y) → (-x, y).
Similarly, the reflection of a point (x, y) over the line x = a is modeled by this transformation rule:
(x, y) → (-2a - x, y)
(x, y) → (-2(-2) - x, y)
(x, y) → (4a - x, y)
By applying a reflection over x = -2 to the coordinate of the given triangle NMO, we have the following coordinates:
(x, y) → (4 - x, y).
N (−5, 2) → (9, 2)
M(−2, 1) → (6, 1)
O(−3 , 3) → (7, 3)
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Answer:
N′(−5, −6), M′(−2, −5), O′(−3, −7)
Step-by-step explanation:
I am in the middle of taking the test and even made a graph to check my work! This should be the correct answer
a school needed a total of $3,546 in Supplies for grade 3,4 and 5 if they spilt the supplies evenly how much did they spend on each grade?
Answer: $1.182
Step-by-step explanation: 3.546 divided by 3
Answer:
$1,182
Step-by-step explanation:
I just took the total of $3546 and divided the total by the three grade levels so 3 and I got the answer 3,546. But honestly i'm not sure if it's right lol, so if it's incorrect let me know and I'll work into it more.
Use a graphing calculator to sketch the graph of the quadratic equation, and then state the domain and range.
y = negative 0.5 x squared + 0.7 x minus 5.1
a.
D: all real numbers
R: (y less-than-or-equal-to negative 5.1)
c.
D: all real numbers
R: (y less-than-or-equal-to negative 4.855)
b.
D: all real numbers
R: (y greater-than-or-equal-to 4.855)
d.
D: all real numbers
R: (y less-than-or-equal-to negative 5.345)
Please select the best answer from the choices provided
A
B
C
D
The domain is all real numbers, and the range is (y ≤ -4.855). (option b)
To sketch the graph of a quadratic equation, you can use a graphing calculator, which is a handy tool that can produce a visual representation of the equation. For instance, let's take the quadratic equation y = -0.5x² + 0.7x - 5.1 and sketch its graph using a graphing calculator.
When we enter this equation into the graphing calculator, we can see a U-shaped curve that is symmetric about the vertical line passing through the vertex. The vertex is the point where the parabola changes direction and is given by the formula x = -b/2a, y = f(x), where f(x) is the value of y at the vertex.
Now let's examine the given options and determine their domain and range based on the graph of the quadratic equation.
all real numbers, R: (y ≤ -4.855)
For this option, we can observe that the parabola opens upward, and the vertex is at (0.7, -5.35).
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Decimal word problem
pls help
Can someone give me a decimal word problem that involves subtraction and division plssss with the answer step by stepp plsss help i begg u guys lovee uuu guyss
Thus, decimal word problem that involves subtraction and division-
The two empty water container weighs 0.64 kg and 1.728 kg when filled with water. Find the water weigh?
Explain about the decimal word problem:The decimal point, which distinguishes between a decimal number's whole number and fractional component, is a little dot. Decimal numbers include 4.14, 3.786, 5.453, and 123.5687 as examples.
Any number of digits can be used to represent the decimal places in a decimal number.The decimal places are the digits that come after the decimal point and are arranged in the following order: tenths, hundredths, thousandths, ten-thousandths, and so forth.decimal word problem : (subtraction and division)
The two empty water container weighs 0.64 kg and 1.728 kg when filled with water. Find the water weigh?
2 water pitcher's weight when empty and full with water is 0.64 kg and 1.728 kg, respectively. We must determine the water's weight. We must deduct the weight of the container when it is empty from the weight of a container when it is full of water in order to determine the weight of the water.Division: Weight of one water container = 0.64 / 2 = 0.34 kg.
Subtraction: Weight of water : 0.728 - 0.34 = 0.385 kg.
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*Time limit help*
Tell me the domain and range
Tell me if the graph is a function or not.
Pick the right answer choice
A. Yes
B. No
C. -2<=x<=6
D. x>=-2
E. All real number
F. x<=6
G. -6<=y<=2
H. y>=-6
I. y<=2
J. y<=6
Examining the graph it can be deduced that
The domain is
C. -2<=x<=6The range is
G. -6<=y<=2The graph is not a function
How to tell if a graph is a function
To qualify as a function, any input (x-value) of a graph must be linked to a single output (y-value). Therefore, each x-value ought to have one and only one y-value that matches it.
A method for identifying whether or not a given graph is representative of a function is by conducting a vertical line test. To achieve this, vertically draw lines through the graph. When every vertical line hits the graph at just one point, then the illustration aptly presents a function. Conversely, if the graph forms multiple intersection points with any vertical line, it isn't classified as a functional representation.
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Given this equation what is the value of y at the indicated point?
We are given the equation of a parabola, [tex]y=-x^2+2[/tex]. The question asks us to find both values of a point given the the x-coordinate which is -3. To find the y-coordinate, simply plug in -3 for x into the equation of the parabola.
[tex]y=-x^2+2 \Longrightarrow y=-(-3)^2+2 \Longrightarrow y=-(9)+2 \Longrightarrow y=-9+2[/tex]
[tex]\Longrightarrow y=-7[/tex]
Thus, the point is (-3,-7)
In parallelogram NPQR if RS=19 find SP.
Answer:
SP = 19
Step-by-step explanation:
the diagonals of a parallelogram bisect each other, then
SP = RS = 19
The graph of y=g(x) is shown. Draw the graph of y=-g(x)+1
The graph of the absolute value function y = -g(x) + 1 is shown in the image attached below.
The graph of the absolute value function y = 2h(x + 4) is shown in the image attached below.
What is a translation?In Mathematics, the translation a geometric figure or graph upward simply means adding a digit to the value on the y-coordinate of the pre-image or function.
In Mathematics, a vertical translation to the positive y-direction (upward) is modeled by this mathematical expression g(x) = f(x) + N.
Where:
N represents an integer.g(x) and f(x) represent a function.In this scenario, we can logically deduce that the equation of the first parent absolute value function is g(x) = |2x| and it would be reflected over the y-axis, and then translated by 1 units upward;
g(x) = |2x|
y = -g(x) + 1 = |2x| + 1
h(x) = |2x + 4| - 4
y = 2h(x + 4)
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Verify the following Pythagorean identity for all values of x and y:
(x² + y²)² = (x² - y²)² + (2xy)²
The identity (x² + y²)² = (x² - y²)² + (2xy)²
What is an identity?An identity is a mathematical equation which shows that one side of the equation is equivalent and equal to the other side of the equation.
To verify the following Pythagorean identity for all values of x and y:
(x² + y²)² = (x² - y²)² + (2xy)², we need to shows that
left-hand side L.H.S = right hand side
So, we proceed as follows
L.H.S = (x² + y²)²
Expanding the brackets, we have that
(x² + y²)² = (x² + y²)(x² + y²)
= (x²)² + 2x²y² + (y²)²
= (x²)² + (y²)² + 2x²y²
Now, we know that (a - b)² = a² + b² - 2ab. So,
a² + b² = (a - b)² + 2ab
Let a = x² and b = y²
So, we have that
(x²)² + (y²)² = (x² - y²)² + 2x²y²
So, substituting this into the equation, we have that
(x²)² + (y²)² + 2x²y² = (x² - y²)² + 2x²y² + 2x²y²
= (x² - y²)² + 4x²y²
= (x² - y²)² + 2²x²y²
= (x² - y²)² + (2xy)²
So, Since L.H.S = R.H.S
(x² + y²)² = (x² - y²)² + (2xy)²
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