The equation does not include the value of 6, we can also express the solution as –[infinity], 6), which excludes the value of 6
The equation 5x - 10 ≤ 20 can be solved by isolating the x variable and rearranging the equation to solve for x. First, we need to add 10 to both sides of the equation in order to get all the terms on one side of the equation. This gives us 5x ≤ 30. Then, we need to divide both sides of the equation by 5 in order to isolate the x variable. This results in x ≤ 6. The solution for this equation can be written as the set of all real numbers less than or equal to 6, which can be represented as –[infinity], 6]. Since this equation does not include the value of 6, we can also express the solution as –[infinity], 6), which excludes the value of 6.
In more detail, when solving a linear inequality, the objective is to isolate the variable on one side of the inequality sign. The first step is to move all terms not containing the variable to the other side of the inequality sign by performing the opposite operation of the given equation. In this case, the equation 5x - 10 ≤ 20 has a subtraction on the left-hand side, so the opposite operation is to add 10 to both sides of the equation. This results in 5x ≤ 30.
Next, we need to divide both sides of the equation by 5 to isolate the x variable. This yields x ≤ 6. The solution for this equation is the set of all real numbers that are less than or equal to 6, which can be written as –[infinity], 6]. Since the equation does not include the value of 6, we can also express the solution as –[infinity], 6), which excludes the value of 6.
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Solve this equation by graphing.First graph the equation then type the solution
Y=1/3x+6
Y=1/6x+5
Answer:
(- 6,4) or x = - 6,y = 4
Step-by-step explanation:
[tex]y=\frac{1}{3}x+6[/tex]
(0,6) and (- 6,4)
(0,5) and (- 6,4)
How does the growth rate of G x compare to the growth rate of f X?
The growth rate of of is smaller than that of of as approaches ∞. Even though the function of equals is growing as increases, it isn't growing as fast as of equals squared. And so the limit of of over of is zero.
Now, According to the question:
Let's know:
What is the growth rate meaning?
The growth rate of a value (GDP, turnover, wages, etc.) measures its change from one period to another (month, quarter, year). It is very generally expressed as a percentage.
The growth rate of of is smaller than that of of as approaches ∞. Even though the function of equals is growing as increases, it isn't growing as fast as of equals squared. And so the limit of of over of is zero.
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PLEASE HELP ASAP - Rewrite the following without an exponent .
(-4)-2 <-- exponent
The value of [tex]-4^{-2}[/tex] without Exponent is [tex]\frac{1}{16}[/tex]
What are Exponents?The exponent of a number indicates how many times a number has been multiplied by itself. For instance, 3^4 indicates that we have multiplied 3 four times. Its full form is 3×3×3×3. Exponent is another name for a number's power. A whole number, fraction, negative number, or decimal are all acceptable.
How many times we must multiply the reciprocal of the base is indicated by a negative exponent. For instance, if a^-n is provided, it can be stretched to 1/a^n. It implies that we must multiply 1/a 'n' times, which is the reciprocal of a. When writing exponentiated fractions, negative exponents are employed.
Calculation:Given;
(-4)⁻² , That implies we have to multiply reciprocal of -4 "2" times .
⇒[tex]\frac{1}{-4}[/tex]×[tex]\frac{1}{-4}[/tex]=[tex]\frac{1}{16}[/tex]
The value of [tex]-4^{-2}[/tex] without Exponent is [tex]\frac{1}{16}[/tex]
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How do we prove that two congruent figures are also similar?
Two congruent figures are similar can be proved by using theorem of similarity given by : SSS, SAS, ASA, AA, and RHS.
Congruent figures are also representing similar figure can be proved in the following ways:
Prove that all the three corresponding sides of the two triangles are in proportion: SSS (Side-side-side).Prove that that corresponding sides of the two triangles are in proportion and included angle is of equal measure : SAS(Side- Angle-Side).Prove that two adjacent angles of one triangle equal to the other triangle: AA ( Angle - Angle).Prove that two adjacent angles of one triangle equal to the other triangle and included side are in proportion : ASA (Angle- Side -Angle)prove that hypotenuse and one of the side of two right angled triangle are in proportion.Therefore, two congruent figures are similar proved by theorem of similarity given by : SSS, SAS, ASA, AA, and RHS.
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Help!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
x= -61
Step-by-step explanation:
[tex]87=-2x-35\\(87)+35=(-2x-35)+35\\122=-2x\\\frac{122}{-2}=\frac{-2x}{-2}\\x=-61[/tex]
What is the median in 1 to 10?
The median of 1 to 10 is 5.5.
The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.
Here we have to find the median of 1-10.
Here we can see that there are 10 numbers given
so, n= 10
Where the number of terms is even.
We know that the formula to find the value of the median whose n value is given,
Median= {(n/2 +1 )th term +(n/2)th term} / 2
We can now substitute the value of n in the above formula, we get
Median= {(10/2+1) +(10/2)} / 2
We can now simplify the above step, we get
Median= 11/2 = 5.5
Therefore, the median of 1 to 10 is 5.5.
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How do you find the x and y intercepts of a logarithmic function?
If there is a y-intercept, we can find it by plugging a zero for x and evaluating the function. If this produces an undefined value, then there is no y-intercept. To find the x-intercept, set y to zero and solve for x.
Now, According to the question:
Logarithmic functions will always have an x-intercept, but they may not have a y-intercept. Let's look at a simple example:
y = [tex]log_1_0x[/tex]
A y-intercept would be located at the y value we get by plugging in a zero for x. The problem is that the function does not have zero as a part of its domain. It is not defined there, so there is no y-intercept. The x-intercept is the x value that causes y to be zero. For a basic logarithmic function like this, that is always x equals 1.
0 = [tex]log_1_0x[/tex]
[tex]10^0[/tex] = x
1 = x
Now we can shift, stretch, and reflect this model to change these results, but the basic idea is still the same. For example, let's consider this model:
y = [tex]log_1_0[/tex] (x + 100) + 2
As always, we look for a y-intercept by plugging in a zero for x. In this case, we do get an answer. This is because of the 100 that is added to the argument of the logarithm.
y = [tex]log_1_0[/tex] (0 + 100) + 2
y = 2 + 2 = 4
As always, to find the x-intercept, we set y to zero and solve for x
y = [tex]log_1_0[/tex] (x + 100) + 2
-2 = [tex]log_1_0[/tex] (x + 100)
[tex]10^-^2[/tex] = x + 100 [By taking antilog on both sides ]
0.01 = x + 100
x = -99.99
So, no matter what the logarithmic function is, we can find the intercepts in this way. If there is a y-intercept, we can find it by plugging a zero for x and evaluating the function. If this produces an undefined value, then there is no y-intercept. To find the x-intercept, set y to zero and solve for x.
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What is the formula for mean median and mode?
Step-by-step explanation:
What is the formula for mean median and mode?
Mean: is the average of values
Median: center most value when values are arranged in numerical order
Mode: most frequent occurring value
_______________________________________
Example
Given values: 0 , 3 , 2 , 10 , 11 , 31 , 2
mean: (0 + 3 + 2 + 10 + 11 + 31 + 2)/7 = 8.429
median: 0 , 2 , 2 , 3 , 10 , 11 , 31
mode: 2 (because 2 shows up twice and the others numbers only once)
What are the roots of 3x² 14x 5 0?
The roots of the quadratic equation 3x²-14x-5 = 0 are 5, -1/3
What is a quadratic equation?The meaning of quadratic equation is any equation containing one term in which the unknown is squared and no term in which it is raised to a higher power.
Given that, a quadratic equation, 3x²-14x-5 = 0
On factorizing, we get,
3x²-14x-5 = 0
3x(x-5)+1(x-5) = 0
(3x+1)(x-5) = 0
(x-5) = 0
x = 5
And,
(3x+1) = 0
x = -1/3
Hence, The roots of the quadratic equation 3x²-14x-5 = 0 are 5, -1/3
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What is the slope for 5?
Using the slope-intercept form, the slope is 0 for line y=5.
What is slope?The slope of a line indicates its steepness. Slope is computed mathematically as "rise over run" (change in y divided by change in x). The slope of a straight line in arithmetic defines how steep it is. It is also known as the gradient. The slope equations. The slope of a line is defined as the "change in y" divided by the "change in x". The slope is defined as the ratio of the vertical change between two places, known as the rise, to the horizontal change between those same two points, known as the run.
Here,
given line,
y=5
y=mx+5
y=0*x+5
m=0
The slope of line y=5 is 0 using the slope-intercept form.
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You’re boarding your plane ride to Austin when you notice that the air
traffic controllers on the ground are
wearing massive earmuffs to prevent ear damage from the plane noise. At what decibel level is sound considered dangerously loud (would cause
permanent ear damage within 30s)?
a. 100 dB
b. 120 dB
c. 150 dB
d. 200 dB
Answer:
Hi! The answer is 120 Db.
Hope it helps!
When bisecting segments and angles, which step is the same?
Draw a ray with one endpoint.
Mark the intersection points of the arcs.
Draw a ray from the vertex to another point.
Place the compass on the endpoint of the segment.
The step that is the same when bisecting segments and angles is "Draw a ray from the vertex to another point."
When bisecting a segment, you must draw a line that divides the segment into two equal parts. One way to do this is to draw a ray with one endpoint at the midpoint of the segment and the other endpoint at any point beyond the segment. This ray will bisect the segment. When bisecting an angle, you must draw a line that divides the angle into two equal angles. One way to do this is to draw a ray from the vertex of the angle to any point on the angle. This ray will bisect the angle.
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Answer: Mark the intersection points of the arcs.
Step-by-step explanation: Draw a ray with one endpoint. is incorrect
Can you prove triangles congruent by SSA?
No, the SSA congruence rule is not a valid criterion for proving whether two triangles are congruent with each other.
What is the SSA Congruence Rule?The SSA Congruence Rule states that two triangles are equal if any two sides and no angle between them are equal to the two sides and the other angle. However, the congruence rule does not say that two triangles are congruent because the sides of the two triangles may not be on the same corresponding side. Both triangles can be different shapes and sizes from each other. Therefore, the SSA Consolidation Rules are not valid. Let's see why SSA doesn't work, see the diagram above. Two triangles ΔABC and ΔDEF have AB = DE, BC = EF, ∠C = ∠F (not included angles). It turns out that two sets of sides and one set of angles are equal (respectively), but the two triangles are not congruent. No two triangles are the same shape and size. Therefore, the SSA Congruence Rule is void.
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What are the five types of domain?
Domains having continuous boundaries, Lipschitz boundaries and C1 boundaries are three types of domain.
Define domain.The collection of all potential inputs for a function is its domain. For instance, the domain of f(x)=x^2 and g(x)=1/x are all real integers with the exception of x=0. Additionally, we can define special functions with more constrained domains.
How to find domain?Consider the function y = f(x), which has the independent variable x and the dependent variable y. A value for x is said to be in the domain of a function f if it successfully allows the production of a single value y using another value for x.
There are three types of domain in mathematics. Domains having continuous boundaries, Lipschitz boundaries, C1 boundaries, and others are frequently thought of as types of domains. An exterior or external domain is the interior of a bounded domain's complement, whereas a bounded domain is a domain that is a bounded set.
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Unit 4
11. Find the minimum, quartile 1, median, quartile 3, and maximum of this dataset:
Wilder 10-day Weather Forecast
78, 83, 86, 86, 74, 69, 72, 76, 78, 79
4
Please help me!!!! With 11!!!!
Find the product and simplify
-2k³ (-3k4 + 5k - 5)
Answer:
The answer is 6k⁷ - 10k⁴ + 10k³
Step-by-step explanation:
-2k³ (-3k⁴ + 5k - 5)
6k⁷ - 10k⁴ + 10k³
Thus, The answer is 6k⁷ - 10k⁴ + 10k³
Maya is playing a trivia game with multiple choice questions. Each question has 2 22 correct answers among 5 55 answer choices. Maya has no idea what the answers to a certain question are, so she needs to choose two different answers at random. What is the probability that maya's first guess is correct and her second guess is incorrect?.
Answer:
0.3 or 3/10
Step-by-step explanation:
First guess:
chance it is right: 2/5
chance it is wrong: 3/5
Second guess, if first is right:
chance it is right: 1/4
chance it is wrong: 3/4
therefore we multipy 2/5 and 3/4 and get a simplified answer of 3/10
Solve 4x4 + 7x2 – 2 = 0. Choose the factored form of the polynomial expression. Select all of the solutions to the equation.
The solutions to the equation are x = ±1/2 and x = √-2 and the factored form is (4x² - 1)(x² + 2) = 0
How to determine the solution to the equation?From the question, we have the following parameters that can be used in our computation:
4x4 + 7x2 – 2 = 0
Rewrite the equation properly
So, we have the following representation
4x⁴ + 7x² – 2 = 0
Expand the equation
This gives
4x⁴ + 8x² - x² – 2 = 0
Factorize the expanded equation
So, we have
4x²(x² + 2) - 1(x² + 2) = 0
Factor out x² + 2
So, we have the following representation
(4x² - 1)(x² + 2) = 0
When the equation is solved, we have
x = ±1/2 and x = √-2
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find the mean of number of books read
The mean of number of students who read fictional books is 4.32.
What is mean?In statistics, the mean refers to the average of a set of values. The mean can be computed in a number of ways, including the simple arithmetic mean (add up the numbers and divide the total by the number of observations).
Given that, the summary of fictional books read by 37 students.
Total number of students =37
Sum =4(2)+10(3)+13(4)+10(7)
= 8+30+52+70
= 160
Now, mean =160/37
= 4.32
Therefore, the mean of number of books read by students is 4.32.
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After observing a swinging lantern, italian scientist galileo galilei made an important discovery about the timing of a pendulum’s swing. christiaan huygens, from holland, discovered the relationship between the length of a pendulum and the time it takes to make a complete swing. these discoveries led to the use of pendulums in clocks.
Galileo's discovery was that the period of swing of a pendulum is independent of its amplitude--the arc of the swing--the isochronism of the pendulum.
Christiaan Huygens, a Dutch mathematician, astronomer, physicist, and horologist, created the pendulum clock in 1656 and received a patent for it in 1657. With the help of this technology, clocks now lose less than 15 seconds every day instead of nearly 15 minutes.
Huygens was motivated by Galileo Galilei's pendulum research, which he began in about 1602. Isochronism, or the fact that a pendulum's period of swing is about the same for swings of all sizes, is the main characteristic that makes pendulums ideal timekeepers, was discovered by Galileo.
Galileo came up with the concept for a pendulum clock in 1637, and his son started building one in 1649, but neither of them survived to see it completed. The precision of clocks was greatly improved by the advent of the pendulum, the first harmonic oscillator used for timekeeping, going from around 15 minutes per day to 15 seconds per day.
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Galileo made the discovery that a pendulum's period of swing is independent of its amplitude, or the swing's arc, or the isochronism of the pendulum.
The pendulum clock was developed in 1656 by Dutch mathematician, astronomer, physicist, and horologist Christiaan Huygens, who also got a patent for it in 1657. With the use of this technology, clocks no longer lose roughly 15 minutes of time each day, but rather lose less than 15 seconds.
Galileo Galilei's pendulum research, which he started about 1602, inspired Huygens. Galileo discovered isochronism, or the fact that a pendulum's period of swing is roughly the same for swings of all sizes, which is the main quality that makes pendulums great timekeepers.
The idea for a pendulum clock was developed by Galileo in 1637, and his son began construction on one in 1649, but neither of them lived to see it finished. With the invention of the pendulum, the first harmonic oscillator used for timekeeping, the accuracy of clocks was substantially increased, decreasing from roughly 15 minutes per day to 15 seconds per day.
Menaha traveled 86km 520m by train and 11km 480m by car What ditance did he travel in all?
In total, Menaha traveled 97km 1000m (97.1km).
What is distance?Distance is a numerical measurement of how far apart two objects, points, or places are in space. Distance can be measured in linear units such as meters, kilometers, feet, miles, etc. It can also be measured in angular units such as degrees or radians.
Distance can also refer to the space between two points in time, such as the time between two events. Distance can be used to measure physical distance, time, or even emotional distance.
To calculate this, the two distances must be added together.
The train distance is =86km 520m (86.52km)
and the car distance is =11km 480m (11.48km).
When added together, =86km 520m+11km 480m = 97.52km.
However, since the distances are measured in km and m,
it is necessary to convert the measurements into a single unit of measurement.
To do this, the measurements must be converted into metres.
The train distance is 86,520 metres
And the car distance is 11,480 metres.
When added together,
the total distance is 97,000 metres (97km).
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Esther and Michael planted a rose in their garden. Esther's rose is x cm tall after 3 weeks. Mike's rose is three less than twice the height of Esther. the sum of their heights decreased by 5 is 22. find the heights of the roses.
HELPPPPPPPP PLEASEEEEEEEEE
The height of the rose of Esther's rose is 10 cm while Mikes's rose is 17 cm.
How to form an equation?Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.
As per the given,
Height of Esther's rose = x cm
Height of Mike's rose = 2x - 3
The sum of their heights decreased by 5 is 22.
x + 2x - 3 - 5 = 22
3x - 8 = 22
x = 10
Thus, Esther's rose = 10 cm and Mike's rose = 2 x 10 - 3 = 17 cm
Hence "Mike's rose is 17 cm tall, compared to Esther's 10 cm high rose".
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The following statement contains an error. Choose the statement that best explains the error.
"The correlation between shoe size and height is 0.87 inches"
A. Correlation requires that both the variables be categorical
B. When stating the correlation coefficient, one must state whether it is a positive or negative relationship
C. This statement does not tell us whether or not shoe size is correlated with height
D. When reporting correlation, one does not report units because correlation has no units
E. There is no error in this statement
The error in the statement will be When reporting correlation, one does not report units because correlation has no units that is option D is correct.
The correct statement would be "The correlation between shoe size and height is 0.87." The correlation coefficient is a measure of the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, with -1 indicating a strong negative relationship, 0 indicating no relationship, and 1 indicating a strong positive relationship. The correlation coefficient does not have units because it is a standardized measure of the relationship between the variables. If there is a unit given in any correlation statement then it means that the statement is a wrong statement or it has an error.
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Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. D is bounded by the parabolas y = x2 and x = y2; p(x, Y) = 19 x
The mass and center of mass of the lamina that occupies the region D and has the given density function p is 57/14 and (14/27, 7/18) respectively.
The center of mass (x―,y―) of a lamina with density function ρ(x,y) is given by
x = M(y)/m, y = M(x)/m
Where, m=∫∫[tex]_{D}[/tex]ρ(x,y)dA
Mx=∫∫[tex]_{D}[/tex] yρ(x,y)dA
My=∫∫[tex]_{D}[/tex] xρ(x,y)dA
Given that, D is bounded by y=x^2 and x=y^2
And ρ(x,y)=19√x
Now, for the point of intersection of y=x^2,x=y^2
we have,
x = (x^2)^2
x = x^4
Subtract x^4 on both side
x - x^4 = 0
x(x^3−1) = 0
x = 0, 1
Now, x=0⇒y=0 and x=1⇒y=1
The points of intersection are (0,0),(1,1)
So, the region D can be written as
D={(x,y): 0≤x≤1, x^2≤y≤x}
So,
m = [tex]\int_{0}^{1}\int_{x^2}^{\sqrt x}19 \sqrt xdydx[/tex]
m = [tex]19\int_{0}^{1}\sqrt{x}[y]^{x^2}_{x}dx[/tex]
m = [tex]19\int_{0}^{1} \sqrt x(\sqrt{x}-x^2)dx[/tex]
m = [tex]19\int^{1}_{0}(x-x^{5/2})dx[/tex]
m = [tex]19[\frac{x^2}{2}-\frac{x^{7/2}}{7/2}]^1_{0}[/tex]
m = [tex]19[\frac{1}{2}(1^2-0)-\frac{2}{7}(1^{7/2}-0)][/tex]
m = 19(1/2−2/7)
m = 57/14
m = 5714
Now,
Mx = [tex]\int_{0}^{1}\int_{x^2}^{\sqrt x}(19xy)dydx[/tex]
Mx = 19[tex]\int^{1}_{0}x[\frac{y^2}{2}]_{x^2}^{\sqrt x}dx[/tex]
Mx = [tex]\frac{19}{2}\int^{1}_{0}x[(\sqrt{x})^2-(x^2)^2}dx[/tex]
Mx = [tex]\frac{19}{2}\int^{1}_{0}(x^2-x^5)dx[/tex]
Mx = [tex]\frac{19}{2}[\frac{x^3}{3}-\frac{x^6}{6}]_{0}^{1}[/tex]
Mx = [tex]\frac{19}{2}[\frac{1}{3}(1^3-0)-\frac{1}{6}(1^6-0)][/tex]
Mx = 19/2 (1/3−1/6)
Mx = 19/12
And
My = [tex]\int^{1}_{0}\int_{x^2}^{\sqrt x}x(19\sqrt{x})dydx[/tex]
My = [tex]19\int_{0}^{1}x^{3/2}[y]^{\sqrt{x}}_{x^2}dx[/tex]
My = [tex]19\int_{0}^{1}x^{3/2}(\sqrt{x}−x^2)dx[/tex]
My = [tex]19\int_{0}^{1}(x^2-x^{7/2})dx[/tex]
My = [tex]19[\frac{x^3}{3}-\frac{x^{9/2}}{9/2}]_{0}^{1}[/tex]
My = 19[1/3(1^3−0)−2/9(1^{9/2}−0)]
My = 19(1/3−2/9)
My = 19/9
So, x = My/m
x = (19/9)/(57/14)
x = (19/9)×(14/57)
x = 14/27
y = Mx/m
y = (19/12)/(57/14)
y = (19/12)×(14/57)
y = 7/18
Therefore, the solutions are (14/27, 7/18).
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The complete question is given below:
What is the mean of 2 3 4 5 0 1 3 3 4 3?
The mean of the data 2, 3, 4, 5,0, 1, 3, 3, 4, 3 is 2.8.
Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.
The basic formula to calculate the mean is calculated based on the given data set. Each term in the data set is considered while evaluating the mean. The general formula for mean is given by the ratio of the sum of all the terms and the total number of terms. Hence, we can say;
Mean = Sum of the Given Data/Total number of Data
Mean= 2 +3 +4 +5 +0 +1 +3 +3 +4 +3 / 7 = 28/10 = 2.8
Hence, The mean of the data 2, 3, 4, 5,0, 1, 3, 3, 4, 3 is 2.8.
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X+2 - x+3;x2+x-2 - x2-1
Answer:
x+2-x+3
x-x+2+3
2+3
5
2x+x-2-2x-1
2x-2x+x-2-1
x-2-1
x-3
What is the pre-image of 5?
Answer : Finding the preimage (s) of a value a by a function f is equivalent to solving equation f(x)=a f ( x ) = a . Finding the preimage (s) of a value a by a function f , which has a known curve, is equivalent to find the abscissae of the intersection(s) of the curve with the ordinate line y=a
In the following list of properties, f: A → B indicates than f is function (or map) from A to B:
If f : A → B then
f [A] = ∅
f -1[B] = ∅.
If f : A → B and X, Y ⊆ B then f-1[X ∩ Y] = f-1[X] ∩ f-1[Y].
If f: A → B and X, Y ⊆ B then f-1[X ∪ Y] = f-1[X] ∪ f-1[Y]
If f: A → B and X, Y, ⊆ A then
f [A ∪ B] = f|A| ∪ f |B|
f [A ∩ B] ⊆ f|A| ∩ f |B|In the following list of properties, f: A → B indicates than f is function (or map) from A to B:
If f : A → B then
f [A] = ∅
f -1[B] = ∅.
If f : A → B and X, Y ⊆ B then f-1[X ∩ Y] = f-1[X] ∩ f-1[Y].
If f: A → B and X, Y ⊆ B then f-1[X ∪ Y] = f-1[X] ∪ f-1[Y]
If f: A → B and X, Y, ⊆ A then
f [A ∪ B] = f|A| ∪ f |B|
f [A ∩ B] ⊆ f|A| ∩ f |B|
The image of 5 is √5.
The preimage of 5 is 25; for ℕ(the set of natural numbers) it is the set of all perfect squares in ℕ [2].
Step-by-step explanation:
What congruent means in math?
Answer:
Answer below
Step-by-step explanation:
The meaning of congruence in Maths is when two figures are similar to each other based on their shape and size.
Answer:exactly equal shape and size
Step-by-step explanation:
What is the equation of this circle in standard form?
Responses
The equation of the circle in standard form from the given graph is
x² + y² + 2x + 2y - 45 = 0
What is a circle?A circle is a two-dimensional figure with a radius and circumference of 2πr
The standard equation of a circle is (x - h)² + (y - k)² = r²
Where (h, k) is the center of the circle
We have,
The standard equation of a circle is (x - h)² + (y - k)² = r²
The coordinates of the center of the circle from the figure.
= (-1, -2)
This means,
(-1, -2) = (h, k)
The radius of the circle is 5√2.
The distance between (-1, -2) and (-6, 3).
= √(-6 + 1)² + (3 + 2)²
= √(25 + 25)
= √50
= 5√2
Now,
The standard equation of a circle is (x - h)² + (y - k)² = r²
(x + 1)² + (y + 2)² = 50
x² + 2x + 1 + y² + 2y + 4 = 50
x² + y² + 2x + 2y - 45 = 0
Thus,
The equation of the circle is x² + y² + 2x + 2y - 45 = 0
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How do you prove I2 =- 1?
We check the validity that i² is equal to -1
The first thing we will do is define numerical sets and complex numbers.
Numerical sets are groupings of numerical values that have a particularity in common, they can be integers, decimals, fractions, among others.
What are Complex Numbers?Among the numerical sets there is one that we call complex numbers, which include values that are not real, such as "i" a letter that denotes that it is an imaginary number.
The definition of a letter value that identifies a type of complex number is the "i" which is the result of the square root of -1, then we have:
√(-1) = i
√(-1) x √(-1) = i²
[√(-1)]² = i²
(-1) = i²
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