Answer:
The coordinate of the midpoint is (\frac{11}{2},\frac{-13}{2} )\\ with endpoints B(9,−8) and C(2,−5)
Step-by-step explanation:
what can prove that
Answer:
Reflecting triangle ABC over line m
Step-by-step explanation:
By reflecting triangle ABC over line m angle A and angle C overlap, proving they are the same angle, which is true because side AB and BC are equal in length.
What is whole number division?
The reverse of multiplying whole numbers is dividing whole numbers. It is a method used to determine the number of times a number (divisor) can be found in another number (dividend).
what is whole number?All natural numbers and 0 are included in a group of numbers known as whole numbers. They belong to the category of real numbers, which excludes fractions, decimals, and negative numbers. Numbers used for counting are also regarded as whole numbers.
Whole numbers are all natural numbers other than zero (0). We are aware that when we talk about natural numbers, we are referring to a group of counting numbers beginning with 1 and continuing through to 9 and beyond. A collection of numbers without fractions, decimals, or even negative integers are known as whole numbers. It is made up of zero and a collection of positive numbers.
The reverse of multiplying whole numbers is dividing whole numbers. It is a method used to determine the number of times a number (divisor) can be found in another number (dividend).
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What are the roots to y x2 4x − 5?
The roots of the Given equation are -5 and 1 solved using the algebraic method to find roots.
The given equation [tex]x^2 + 4*x-5[/tex] is a quadratic equation.
how to identify whether a given equation is a quadratic equation or not?
The polynomial equations of degree two in one variable of type [tex]f(x) = ax^2 + bx + c = 0[/tex] and with a, b, c, belongs to Real Number and a not equal to 0 is known as a quadratic equation. It is a quadratic equation in its general form, where "a" stands for the leading coefficient and "c" is for the absolute term of f. (x). The roots of the quadratic equation are the values of x that satisfy the equation.
x^2 + 4*x -5 = 0
x^2 + (5-1)*x - 5 = 0
x^2 + 5*x - x -5 = 0
x(x+5) - 1(x+5) = 0
(x+5)(x-1) = 0
if a*b = 0 then a = 0 or b= 0.
In the same way
x+5 = 0 or x - 1 = 0
x = -5 or x = 1.
So Roots of the Given equation are -5 and 1.
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What is value with example?
Value is the worth of something in terms of the amount of other things it can be exchanged for. For example, a car is valued at $20,000, meaning that it can be exchanged for $20,000 worth of goods or services.
The Value of Something: How Do We Determine Its Worth?When trying to determine the value of something, it is important to consider its worth in terms of the amount of other things it can be exchanged for. This is because the value of an item is defined as the amount of other goods or services that it can be exchanged for. In other words, it is the amount of money that can be obtained from trading it.
For example, a car is typically valued at $20,000, meaning that it can be exchanged for $20,000 worth of goods or services. However, it is also important to consider the condition of the car and other factors that may influence its value. For instance, if the car has a lot of wear and tear, its value may be lower than $20,000. On the other hand, if the car is in excellent condition, its value may be higher than $20,000.
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A landscaper wishes to plant a uniform border of tulips within a
rectangular garden with dimensions 18 m by 12 m. To obtain a
pleasing look the area of the tulip border should be half the area
of the garden. How wide should the border be, to 1 decimal
place?
The tulip border should be half the size of the garden, which is 108m.
Explain about the rectangular?A rectangle is a closed 2-D shape with four sides, four corners, and four right angles (90°). The opposing sides of a rectangle are equal and parallel. A rectangle has two dimensions: length and width, as it is a two-dimensional structure.
A rectangle is a type of quadrilateral with four 90-degree-distance vertices and parallel sides that are equal to one another. It is also referred to as an equiangular quadrilateral because of this. A rectangle can also be referred to as a parallelogram since its opposing sides are equal and parallel.
A quadrilateral with four equal and right-angled angles is a rectangle. A square also has all equal sides in addition to all equal angles.
Tulip border area = (1/2) (area of garden)
= (1/2) (18 x 12)
= (1/2) (216)
= 108
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Is Y 3x inverse or direct?
y = 3x is direct variation, not an inverse variation.
What is inverse and direct variation?When x is not equal to zero, an equation of the form y = kx describes the linear function known as direct variation. An equation of the form xy = k, where x is not equal to zero and k is a nonzero real number constant, describes the nonlinear function known as inverse variation.
Direct variation describes how one quantity directly affects the other when it varies. The precise reverse of this is called inverse variation.
The value of the proportionality constant, k, may be expressed as k = y/x if x is not equal to zero. As a result, the ratio between these two variables is always a fixed value. The direct variation equation may also be written as x = y / k.
In this case y = 3x is equals to y = kx.
So, it is direct variation.
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The complete question is as follows:
Is Y = 3x inverse or direct?
If PQRS ~ WXYD, find the value of x.
(If possible, show step-to-step)
Answer:
x = 60
Step-by-step explanation:
We are told that the figures are similar. That means the that corresponding sides are proportional.
Sides Ps and ZW are corresponding sides. We can use them to find the scale factor form the smaller figure to the larger figure.
40 times some number (the scale factor) is 75
Let x = the scale factor
40x = 75 Divide both sides by 40
[tex]\frac{40x}{40}[/tex] = [tex]\frac{75}{40}[/tex]
x = 1.875 This is are scale factor.
Take side YZ and multiply its length of 32 by the scale factor to find x.
32(1.875) = 60
Answer:
x = 60
Step-by-step explanation:
assuming you mean PQRS is similar to WXYZ , then ratios of corresponding sides are in proportion, that is
[tex]\frac{RS}{YZ}[/tex] = [tex]\frac{PS}{WZ}[/tex] ( substitute values )
[tex]\frac{x}{32}[/tex] = [tex]\frac{75}{40}[/tex] ( cross- multiply )
40x = 2400 ( divide both sides by 40 )
x = 60
How do you know if a graph is linear?
An equation that depicts a straight line on a graph is said to be linear line.
What is linear line?Linear refers to a straight line. Straight-line graphs called linear graphs are used to show the relationship between two values. The single straight lines on this graph assist to represent a result. The phrase "linear" refers to a straight line and excludes the usage of curves, dots, bars, etc.
A linear line can include more than one variable. Linear lines with two variables, for example, are those that have two variables in them. 2x - 3 = 0 and 2y = 8 are few examples of linear lines.
Steps to draw a linear graph are described below:
A linear relation's graph can be discovered by graphing at least two points. The intercepts lead to the discovery of two locations that are very helpful for illustrating the graph of a line.A graph's x-value at which it crosses the x-axis is known as the x-intercept. A graph's y-value at where it crosses the y-axis is known as a y-intercept.Since the x-axis has a value of y = 0, an x-intercept may be discovered by setting y in the equation to 0 and figuring out x.Setting x = 0 in the equation and calculating y leads to the discovery of a -y intercept.Examine the line to see if it is straight or curved. A straight line indicates a linear equation. A nonlinear equation is one that is curved.
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What does end behavior tell you about the graph?
The end behavior of a function f describes the behavior of the graph of the function at the "ends" of the x-axis. In other words, the end behavior of a function describes the trend of the graph
What is the End Behavior of a Function?
When a function f (x) increases or decreases without bound, the behaviour at its finish is referred to as the function's behaviour. increases or decreases without bound. In other words, the end behavior describes the ultimate trend in the graph of f(x) as we move towards the far right or far left of the x-axis.
End behavior is represented in mathematical notation by symbols that indicate the impact on the function when the variable goes toward plus or negative infinity. For instance, the claim
The expression f(x)+ as x+ indicates that the function will grow unboundedly huge as the variable rises and can be read as "f(x) approaches positive infinity as x approaches positive infinity."
Because it can be considered as defining the long-term behavior of a quantity of interest, the concept of the end behavior of a function has practical applications. For instance, epidemiologists might be interested in predicting future trends in the prevalence of a disease. Whether or not case numbers are expected to plateau or keep rising is a sign of whether or not the epidemic is currently under control.
so ,The end behavior of a function f describes the behavior of the graph of the function at the "ends" of the x-axis.
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The behavior of the function graph at the "ends" of the x-axis is known as the end behavior of a function, or f. In other words, the graph's trend is described by the function's final behavior.
The formula f(x)+ as x+, which may be translated as "f(x) approaches positive infinity as x approaches positive infinity," denotes that the function will become unboundedly massive as the variable increases.
so ,
The behavior of the function graph at the "ends" of the x-axis is known as the end behavior of a function, or f.
What is the mean in math calculator?
A dataset's mean (also known as the arithmetic mean, which differs from the geometric mean) is calculated by dividing the sum of all values by the total number of values. The term "average" is frequently used to describe this measure of central tendency.
What does the mode mean?
By summing all the numbers in a data collection and dividing by the total number of values in the set, the mean (average) of the data set can be determined.
A data collection is ordered from least to largest, and the median is the midpoint number. A data set's mode is the number that appears the most frequently.
What are the mean, median, and example?
The most frequent number, or the one that happens the most frequently, is known as the mode.
Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.
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How is 2 a rectangular number?
Answer:
It is one row by two columns.
Step-by-step explanation:
What type of solution does the pair of equation 2x 3y 8 4x 6y 9 have?
The equations cannot be solved since they are parallel to one another. The equations are hence incoherent.
How many solutions does the pair 2x 3y 8 4x 6y 9?The supplied equations are 2x3y=7, 4x6y=20, 4x6y=14, and so on. Therefore, the first equation and the second equation are in conflict. As a result, these equations will not have a solution since no value for x or y will satisfy both equations.
The junction of two lines at the spots where a linear equation has a solution are known as the points of intersection. The collection of all feasible values for the variables that satisfy the given linear equation is, in other words, the solution set of the system of linear equations.
There is no solution to the system of linear equations if the graphs of the equations are parallel. No point can be found in this situation when no lines cross one another.
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Ana drew the parent graph of y=x?. How should she transform that graph to produce the graph of y=2(×+ 7)square
For the parent function of y = x as sketched by Ana the transformation to get the equation y = 2(x + 7 ) is
translate to the left 7 units
vertical stretch by a factor of 2
What is transformation?A point, line, or geometric figure can be transformed in one of four ways, each of which affects the shape and/or location of the object.
Pre-Image refers to the object's initial shape, and Image, after transformation, refers to the object's ultimate shape and location.
The type of transformation involved here is translation and vertical stretch
form the parent function, y = x
translation of seven unites to the left results to y = x + 7
vertical stretch of 2 units results to y = 2(x + 7)
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6. A shop keeper has 56 pens. He
soid them for 280 dollars. How
much did the shopkeeper charge
for each pen?
Answer:
5
Step-by-step explanation:
280 divided by 56 = 5
to check your answer just 5x56=280
hope this helped
What is the standard deviation of 16 24 18 17 12 17 25 15 19 20 27 24?
Standard deviation (S.D) and Mean are 14.603 and 19.5 respectively.
Given data
16, 24, 18, 17, 12, 17, 25, 15, 19, 20, 27, 24
here we need to find standard deviation of the given data from mean
standard deviation of a data = ∑x²/ N
here ∑x² = sum of squares of deviations of the data
N = number of observations
⇒ Mean of the data = [16+24+18+17+12+17+25+15+19+20+27+24]/ 12
= 234 / 12 = 19.5
⇒ now calculate deviations in following table
observations deviations squares of deviations
16 3.5 12.25
24 4.5 20.25
18 1.5 2.25
17 2.5 6.25
12 7.5 56.25
17 2.5 6.25
25 5.5 30.25
15 4.5 20.25
19 0.5 0.25
20 0.5 0.25
27 0.7 0.49
24 4.5 20.25
N = 12 ∑ x² = 175.24
standard deviation (S.D) = 175.24/ 12 = 14.603
Hence Standard deviation (S.D) and Mean are 14.603 and 19.5 respectively.
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An equation is given. Part A. x2 = 49 What are the values of x? Part B. Solve for b in the equation -64 = b³
Answer:
Step-by-step explanation:
Joscelyn, we know that [tex]x^{2}[/tex] = 49 means ,
x = [tex]\sqrt{49}[/tex]
7 * 7 = 49
x = 7
for b) it looks like the question is
-64 = [tex]b^{2}[/tex]
teacher don't like to tell anyone that you can have a negative under the square root sign. Because it means you have to understand imaginary numbers. :/
[tex]\sqrt{-64}[/tex] = b
another way to write the negative sign is [tex]i^{2}[/tex]
[tex]\sqrt{64i^{2} }[/tex] = b
use properties of square roots
[tex]\sqrt{64}[/tex]*[tex]\sqrt{i^{2} }[/tex] = b
8i = b
written well, this looks like
0 + 8i = b
if the questions doesn't have the negative sign
then it's just
8 = b
The value of x = +7, -7 and the value of b = -4
Given an equation :
x² = 49,
b³ = -64
x * x = 49 ( 7 * 7 = 49 and -7 * -7 = 49 )
b * b * b = -64 ( (-4) * (-4) * (-4) )
On solving
x = +7, -7 and b = -4
An equation is a condition on a variable while an expression is simply a polynomial in one or more variables.
An equation contains an equal sign while an expression does not contain an equal sign. An expression contains operators, constants, and variables in it.
answer x = +7, -7 and b = -4
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Jose has a test coming up. On his previous 4 tests he had scores of 92, 74, 88, and 85. What must he
score on this fifth test to have an overall average of 90? Is this possible?
Answer:
111
Step-by-step explanation:
Add up the values of the four tests 92+74+88+85=339
Then 90 x 5=450 for five test
Find the difference between the two numbers
450-339=111
Show that 1/x is an integrating factor of the equation dy/dx-y/x = 3x
Suppose we're asked to solve the first order linear differnetial equation,
[tex]\longrightarrow\rm{\dfrac{dy}{dx}+y\cdot P(x)=Q(x)}[/tex]
To solve this equation we have to find out a term called 'Integrating Factor' given by,
[tex]\rm{I_f=e^{\int P(x)\ dx}}[/tex]
such that the solution of the equation is,
[tex]\displaystyle\longrightarrow\rm{y\cdot I_f=\int Q(x)\cdot I_f\ dx}[/tex]
Here the given differential equation is,
[tex]\longrightarrow\rm{\dfrac{dy}{dx}-\dfrac{y}{x}=3x}[/tex]
[tex]\longrightarrow\rm{\dfrac{dy}{dx}+y\left(-\dfrac{1}{x}\right)=(3x)}[/tex]
Here,
[tex]\longrightarrow\rm{P(x)=-\dfrac{1}{x}}[/tex]
Integrating wrt x,
[tex]\displaystyle\longrightarrow\rm{\int P(x)\ dx=-\int\dfrac{1}{x}\ dx}[/tex]
[tex]\displaystyle\longrightarrow\rm{\int P(x)\ dx=-\ln|x|}[/tex]
[We will consider constant of integration as zero.]
Then the integrating factor is,
[tex]\longrightarrow\rm{I_f=e^{\int P(x)\ dx}}[/tex]
[tex]\longrightarrow\rm{I_f=e^{-\ln|x|}}[/tex]
Since [tex]\rm{e^{b\,\ln(a)}=a^b,}[/tex]
[tex]\longrightarrow\rm{I_f=x^{-1}}[/tex]
[tex]\longrightarrow\rm{\underline{\underline{I_f=\dfrac{1}{x}}}}[/tex]
Hence the integrating factor for the given differential equation is 1/x.
At Keller's Bike Rentals, it costs $16 to rent a bike for 4 hours. How many hours of bike use does a customer get per dollar?
Answer:
.25 hours
Step-by-step explanation:
if its $16 to rent a bike for 4 hours, each hour costs 4 dollars meaning that one dollar would be .25 of an hour
Answer: 15 minutes
Step-by-step explanation:
First, take 16 divided by 4 = $4 per hour
We know that 1 hour = 60 minutes
Take 60 divided by 4 = 15 minutes per dollar
15 minutes = 0.25 hour
In 2020, the student population of a small school is 284. It is estimated that it will increase 4% each year.
Estimate the student population in 2026.
2149450.133 is the formula for the anticipated student population in 2026.
What is meant by population?A population is a distinct group of people, animals, or other objects that can be identified for the purposes of data collection and analysis by at least one common trait. The majority of the time, data is collected from samples in order to learn more about huge populations. the entire population being investigated. Example: At a football game, you ask 100 people chosen at random what their primary occupation is. 100 persons make up your sample, but everyone present at the match is the population.
Given:
Population in 2020: 284
Population growth annually is 4%.
Year predicted is 2026.
Let n = the difference in population between the forecasted year and 2020 (2026 minus 2020 = 6).
The equation:
Estimated population is equal to the sum of the population in 2020 and the annual growth rate.
= 284 + (284 × 4%)ⁿ
= 284 + (284 × 0.04)⁶
= 284 + (11.36)⁶
= 284 +2149166.133
=2149450.133
Therefore, the estimated population of the students in 2026 is 2149450.133.
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How do you write less than 70?
We can write 'less than 70' as < 70, using the less-than sign.
The less-than sign is a mathematical symbol that denotes an inequality between two values. The widely adopted form of two equal-length strokes connecting in an acute angle at the left, <, has been found in documents dated as far back as the 1560s. In mathematical writing, the less-than sign is typically placed between two values being compared and signifies that the first number is less than the second number. Examples of typical usage include 1⁄2 < 1 and −2 < 0.
Since the development of computer programming languages, the less-than sign and the greater-than sign have been repurposed for a range of uses and operations.
We can use this mathematical symbol to write "less than 70" in a mathematical form.
< 70, indicating that the number is less than 70.
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Write the slope intercept form of the line below:
Through (-3,1) parallel to y=-4/3x-1
and..
Through (-3,-5) perpendicular to x+2y=-4
The slope-intercept form of the line through (-3,1) parallel to y=-4/3x-1 is y = (-1/2)x - 2 while through (-3,-5) perpendicular to x+2y=-4 is y = 2x + 1.
What is a linear function?A straight line on the coordinate plane is represented by a linear function.
A linear function always has the same and constant slope.
The formula for a linear function is f(x) = ax + b, where a and b are real values.
As per the given,
The slope of y = -4/3 x - 1 is -4/3.
Since parallel lines have the same slope thus,
y = (-4/3)x + c
Put (-3,1) into the above equation,
1 = (-4/3)(-3) + c
c = -3
Thus, equation will be y = (-4/3)x - 3
As per the given perpendicular line,
x+2y=-4
y = (-1/2)x - 2
Slope = 2
Put (-3,-5) into the above equation,
-5 = 2(-3) + c
c = 1
Thus, the equation will be y = 2x + 1
Hence "The line through (-3,1) parallel to y=-4/3x-1 has a slope-intercept form of y = (-1/2)x - 2, while the line through (-3,-5) perpendicular to x+2y=-4 has a slope-intercept form of y = 2x + 1".
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What number is 5% of 20?
Answer:
Step-by-step explanation:
1
Answer: 5% of 20 is 1.
How do you determine the nature of the roots of a quadratic equation?
To determine the properties of the roots of the quadratic equation (of the form ax^2 + bx +c=0), we need to compute the discriminant (b^2 - 4 a c). If the discriminant is greater than 0, the roots are unequal and real. If the discriminant is zero, the roots are equally real.
In algebra, the quadratic equation (from the Latin quadratus 'square') is rewritten in standard form as ax² + bx +c = 0
where x represents an unknown value, a, b, and c represent known numbers and a ≠ 0. (If a = 0 and b ≠ 0, the equation is linear rather than quadratic.) The numbers a, b, and c are the coefficients of the equation, the quadratic coefficient, the linear coefficient, and the constant or free period and called.
For the quadratic equation ax²+ bx+ c=0, a ≠ 0 , we found that the roots of that equation are given by
[tex]\alpha = \frac{-b+\sqrt{b^2-4ac} }{2a}[/tex] and [tex]\beta = \frac{-b-\sqrt{b^2-4ac} }{2a}[/tex]
The term inside the square root is called the discriminant. Discriminants can be used to analyze the types of solutions to quadratic equations without actually solving the equations. Method is as follows.
D = b²-4ac > 0
If an equation has two distinct real solutions, i.e. α and β are real.
α - β = [tex](\frac{-b+\sqrt{D} }{2a} ) -( \frac{-b - \sqrt{D} }{2a})[/tex]
= [tex]\frac{-b +\sqrt{ D } +b + \sqrt{D} }{2a}[/tex] = 2√D/2a= √D/a
i.e., α-β ≠0
⇒ α ≠ β
i. For perfect squares, the roots are real, rational, and unique.
ii. If not a perfect square, the root is real, irrational, and unique.
if D = b²- 4ac =0, the equation has a real solution, i.e. a double root. Also, the α = β = -b/2a
If D= b²- 4ac < 0, then equation has only non-real solutions.
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(0,1); (2,-4) what is the slope?
Step-by-step explanation:
the answer to your problem is m= -5/2
Answer:
-2.5
Step-by-step explanation:
The slope formula [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex] tells us the formula for any slope with 2 given points. We have to plug in the numbers from our 2 ordered pairs to get [tex]\frac{-4 - 1}{2 - 0}[/tex], which simplifies to [tex]\frac{-5}{2}[/tex] and can be written as -2.5
The slope of this equation is -2.5
What is the nature of roots of quadratic equation x² =- 2x 1?
The nature of roots of quadratic equation x²-2x-1 are real and distinct.
What is quadratic equation?A quadratic equation is a second-order polynomial equation in a single variable x , ax²+bx+c=0. with a ≠ 0 .
The given quadratic equation is x²-2x-1=0.
Nature of Roots depending upon Discriminant
If the discriminant is equal to zero (b² – 4ac = 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax² + bx + c = 0, are real and equal.
a=1, b=-2 and c=-1
b² – 4ac =4-4(1)(-1)
=4+4=8>0
b² – 4ac>0
So roots are real and distinct.
Hence, the nature of roots of quadratic equation x²-2x-1 are real and distinct.
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I need help with this
7x2 −5x 2 −40x−30
What is f(x)?
(7x 3−5x+42x−30)+(7x−5)
Add the terms
7x 3+37x−30+(7x−5)
Subtract the terms
7x 3+37x−30+7x−5
Collect like terms
7x 3 +44x−30−5
Solution
7x 3+44x−35
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If θ=7π/6, then
tan(θ)=
cot(θ)=
sec(θ)=
csc(θ)=
Give exact values. No decimals allowed!
The exact values of the trigonometric functions are:
tan (7π / 6) = √3 / 3
cot (7π / 6) = √3
sec (7π / 6) = - 2√3 / 3
csc (7π / 6) = - 2
How to find the exact values of trigonometric functions
The angle given in the statement can be represented in a goniometric circle by a ordered pair of the form (x, y) = (cos θ, sin θ) and the expressions of the derivated trigonometric functions can be found by means of the following definitions:
Tangent
tan θ = sin θ / cos θ
Cotangent
cot θ = cos θ / sin θ
Secant
sec θ = 1 / cos θ
Cosecant
csc θ = 1 / sin θ
First, find the exact values of the sine and cosine of the angle:
sin (7π / 6) = sin (π + π / 6) = sin π · cos (π / 6) + cos π · sin (π / 6)
sin (7π / 6) = - sin (π / 6)
sin (7π / 6) = - 1 / 2
cos (7π / 6) = cos (π + π / 6) = cos π · cos (π / 6) + sin π · sin (π / 6)
cos (7π / 6) = - cos (π / 6)
cos (7π / 6) = - √3 / 2
Second, calculate the exact values of the derivate trigonometric functions:
tan (7π / 6) = (- 1 / 2) / (- √3 / 2)
tan (7π / 6) = √3 / 3
cot (7π / 6) = √3
sec (7π / 6) = 1 / (- √3 / 2)
sec (7π / 6) = - 2√3 / 3
csc (7π / 6) = 1 / (- 1 / 2)
csc (7π / 6) = - 2
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What is the nature of the roots of the quadratic equation 3x² 4x 2?
The nature of the roots of the quadratic equation 3x² -4x- 2 has two distinct real roots.
What is quadratic equation?A quadratic equation is a second-order polynomial equation in a single variable x , ax2+bx+c=0. with a ≠ 0 .
The given quadratic equation is 3x² -4x- 2.
Three times of x square minus four times of x minus two.
Nature of Roots depending upon Discriminant.
If the discriminant is equal to zero (b² – 4ac = 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax² + bx + c = 0, are real and equal.
b=-4, c=-2, a=3
b² – 4ac=16-4(3)(-2)
=16+24=40
b² – 4ac>0
The quadratic function has two distinct real roots.
Hence, the nature of the roots of the quadratic equation 3x² -4x- 2 has two distinct real roots.
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What are all the roots of
x^3 +2x = x^2 +2?
Answer:
Step-by-step explanation:
To find the roots of the equation x^3 + 2x = x^2 + 2, we need to solve the equation for x. To do this, we can start by subtracting x^2 and 2 from both sides:
x^3 + 2x - x^2 - 2 = 0
This simplifies to x^3 - x^2 + 2x - 2 = 0. Next, we can factor the left side of the equation as follows:
(x - 2)(x^2 + 2x + 1) = 0
Since the product of two factors is equal to 0, this means that at least one of the factors must be equal to 0. Therefore, the roots of the equation are the solutions to the equations x - 2 = 0 and x^2 + 2x + 1 = 0. The first equation has a single solution at x = 2, and the second equation is a quadratic equation that can be solved using the quadratic formula:
x = (-2 +/- sqrt(4 - 411)) / (2*1)
= (-2 +/- sqrt(4 - 4)) / 2
= (-2 +/- sqrt(0)) / 2
= (-2 +/- 0) / 2
= -1 or -1
Therefore, the roots of the equation x^3 + 2x = x^2 + 2 are x = 2, x = -1, and x = -1.