First make the graph of the functions, then if the two graphs are symmetric with respect to the line y = x (mirror images over y = x ), then they are inverse functions.
Symmetric
A figure or shape that can be divided into two equal parts by a line is called symmetric figures.
Inverse Functions
The inverse function of a function f is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by F^-1.
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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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Add.
6 38/45+2 5/9
Enter your answer in the box as a mixed number in simplest form.
Answer:
28 1/5
Step-by-step explanation:
6 38/45+2 5/9
6 38/45 = 308/45
2 5/9 = 23/9
The common factor is 45
308/45 + 115/45 = 423/15 = 141/5
141/5 = 28 1/5
Answer:
9[tex]\frac{2}{5}[/tex]
Step-by-step explanation:
6 [tex]\frac{38}{45}[/tex] + 2[tex]\frac{5}{9}[/tex] Use a common denominator of 45 so that you can add together
6 [tex]\frac{38}{45}[/tex] + 2[tex]\frac{25}{45}[/tex] Multiply the top and bottom of the second fraction by 5
8 [tex]\frac{63}{45}[/tex]
8 + [tex]\frac{45}{45}[/tex] + [tex]\frac{18}{45}[/tex] I broke up [tex]\frac{63}{45}[/tex] because I know that [tex]\frac{45}{45}[/tex] is another name for 1
8 + 1 + [tex]\frac{18}{45}[/tex]
9 [tex]\frac{18}{45}[/tex] Divide the numerator and the denominator by 9 to simplify
9[tex]\frac{2}{5}[/tex]
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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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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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?
What number is 5% of 20?
Answer:
Step-by-step explanation:
1
Answer: 5% of 20 is 1.
There are 36 kids in the class.
1/4
of the kids are teenagers.
1/3
of the teenagers are wearing leggings.
How many kids in the class are teenagers wearing leggings?
The number of kids in the class teenagers wearing leggings is 3.
What is a fraction?In such a fraction, the value above the horizontal line is referred to as the numerator.
In another word, if we divide two numbers by each other then the upper word is called the numerator, and the lower word is called the denominator.
As per the given,
Number of kids = 36
Fraction of kids are teenagers = 1/4 of 36
(1/4)36 = 9 kids
Fraction of teenagers wear leggings = 1/3 of 9
(1/3)9 = 3 kids
Hence "There are three students in the class who are teenagers and wearing leggings".
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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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What are the vertical and horizontal asymptotes of F x )= 3x 2 x 2 4?
The horizontal asymptote of F(x)=3x^2/x^2-4 is y=0, meaning the graph approaches this line as x goes towards infinity. The vertical asymptote is x=2, which is a double root of the denominator; this means the graph approaches a line parallel to the x-axis as x approaches 2.
The horizontal asymptote of F(x)=3x^2/x^2-4 is y=0. This means that as x goes towards infinity, the graph approaches a line parallel to the y-axis. The vertical asymptote is x=2, which is a double root of the denominator; this means the graph approaches a line parallel to the x-axis as x approaches 2. In other words, the graph has a vertical line at x=2 where the graph approaches, but never reaches, a horizontal line. The fact that the vertical asymptote is a double root of the denominator means that the graph will approach two lines, one on either side of the asymptote, as x approaches 2. It is important to note that the vertical asymptote is not the same as the y-intercept, which will be located at the point where the graph intersects the y-axis.
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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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What is stated in rational root theorem?
The rational root theorem states that it is a representation to determine the rational zeros or the solution of the given polynomial function.
As given in the question,
Rational zero theorem is also known as the rational solution or root theorem.we can suppose one of the polynomial function h(x) = a₀ + a₁x + a₂x² + ...+ aₙxⁿRational root theorem or the solution which can be find out in the simplest form m/n by using the derived formulam / n = ( One of the factor of a₀ ) / ( One of the factor of aₙ )Value of m/n which was reduced to simplest form is always a prime number.They are also known by the name of root, zeros or the solution of the any polynomial function.Therefore, the rational root theorem helps us to find rational solution or zeros of the of any polynomial function.
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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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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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Define variables for number of 4 1/2 feet pieces of rope and the amount of the 48 feet long rope that’s left over then write an equation to represent the real ashton’s hip between these variables.
9/2 x +48 is an equation to represent the real ashton’s hip between these variables.
To write this type of fraction = 4 + 1/2 - The integer 4 in the expression is the amount of feet of rope that was cut.
Thus to know the number of 1/2 foot pieces produced, let this number be x. Thus, we have 1/2 of x pieces = 2
x/2 = 4; x = 2x4 = 8
Thus, it has 8 pieces of 1/2 foot long piece each of rope.
9/2 x +48 is an equation to represent the real ashton’s hip between these variables.
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Hot dogs come 8 to a package. Buns come 6 to a package. What is the fewest number of packages of each you would have to buy so that you have exactly as many hot dogs as buns?
____packages of hot dogs; _____packages of buns
Answer:
3 hot dogs, 4 buns
Step-by-step explanation:
You can figure this out
Answer: 3 packages of hot dogs, and 4 packages of buns
Step-by-step explanation:
8 x 3 =24
6 x 4 =24
same amount of hot dogs as there are buns
HELP EMERGENCY ITS DUE TODAY 4100 people attended a football game. If 20% of the people who attended were teenagers, how many teenagers attended the game?
Answer:
820 teens
Step-by-step explanation:
20 percent is also 1/5 so 4100*1/5 is 4100/5 so it is 820
What are the factors of 3x² 4x 4?
The factors of 3[tex]x^{2}[/tex] -4x -4 is (3x+2)(x-2)
Let's start by putting empty brackets.
()
We know that 3[tex]x^{2}[/tex] is 3x*x
Each of these can be inserted into a particular bracket.
(3x)(x)
The factors of -4 are then examined. How many different methods are there to multiply two numbers to reach -4?
1 × -4 = -4
2 × -2 = -4
Choosing which pair to utilize and where to place each is now the challenging part. You might need to try each one separately at first, but with practice this will become simpler. I'll begin with the right couple, which are +2 and -2.
(3x + 2)(x - 2)
We can enlarge it once again to confirm whether this is accurate. Do not forget to multiply each value in the second bracket by each value in the first bracket
so, 3x *x = 3[tex]x^{2}[/tex]
3x * -2 = -6x
Then
2 * x =2x
2 * -2 = -4
All of these together gives us 3[tex]x^{2}[/tex] - 6x +2x -4 which is 3[tex]x^{2}[/tex] - 4x -4
If we had reversed the factors by putting (3x−2)(x+2), it would have given us 3[tex]x^{2}[/tex] - 4x -4, which is close but not exactly what we are looking for.
If the alternative set of components, 1 and -4, had been employed, we would have had ( 3 x + 1 ) ( x- 4 ), which would be 3[tex]x^{2}[/tex]- 11x -4
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How is 2 a rectangular number?
Answer:
It is one row by two columns.
Step-by-step explanation:
What is the area of the base square?
Base area can be determined using the same method as the square's area, which is side side or base edge².
Define the term area of base square?A square pyramid consists of three parts. Apex refers to the pyramid's highest point. The base is the square at the bottom. Faces are the three triangle sides.
A square pyramid consists of three parts.
Apex refers to the pyramid's highest point.The base is the square at the bottom.Faces are the three triangle sides.Square Pyramid Base Area
Due to the square base of the square pyramid, its base area can be determined using the same method as the square's area, which is side side or base edge².
Example: Assume that a square pyramid's base edge is given as seven units. Thus, BA = 7 x 7 = 49 square units is the size of the square pyramid's base.
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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.
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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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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can any one help in these questions
Answer:
1) x = 10
m<LOH = 55
m<GOH = 85
m< KOG = 40
2) x = 3
m<CQR = 22
m<BQC = 53
m< AQB = 33
m< PQA = 72
3) x = 45
m<TYZ = 41
m<XYT = 49
m< SYX = 90
m< XYZ = 90
4) x = 7
m < KJL = 30
m< NJK = 50
m< MJL = 56
Step-by-step explanation:
1)
Straight line = 180°
So
4x + 15 + 9x - 15 + 4x = 180
17x + 10 = 180
x = (180-10)/17
x = 10
m<LOH = 40
m<GOH = 85
m< KOG = 55
2)
Straight line = 180°
So
7x + 1 + 56 - x + x + 30 + 24x = 180
31x + 87 = 180
x = (180 - 87) / 31
x = 3
m<CQR = 72
m<BQC = 33
m< AQB = 53
m< PQA = 22
3)
Straight line = 180°
So
2x + x + 4 + x - 4 = 180
4x = 180
x = 45
m<TYZ = 41
m<XYT = 49
m< SYX = 90
m< XYZ = 90
4)
Straight line = 180°
So
2(3x + 4) + 11x - 3 + 8x = 180
6x + 8 + 19x -3 = 180
25x + 5 = 180
x = (180 -5) / 25
x = 7
m < KJL = 30
m< NJK = 50
m< MJL = 56
I hope my answer helps you.
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.
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
According to the rational root theorem what are all the potential rational roots of f(x)=15x^11-6x^8.
The potential rational roots must be factors of -6/15. These factors are ±1, ±2, ±3, ±6, and ±10.
The rational root theorem states that all rational roots of a polynomial equation must be factors of the constant term divided by factors of the leading coefficient.
In the case of the polynomial f(x)=15x^11-6x^8, the constant term is -6 and the leading coefficient is 15. This means that all potential rational roots must be factors of -6/15. These factors are ±1, ±2, ±3, ±6, and ±10. Therefore, the potential rational roots of f(x)=15x^11-6x^8 are ±1, ±2, ±3, ±6, and ±10.
The rational root theorem is an important tool for solving polynomial equations. It allows us to quickly determine all the potential rational roots of a polynomial equation, which can then be used to solve the equation. In the case of f(x)=15x^11-6x^8, the potential rational roots can be used to determine the factorization of the polynomial. For example, if we were to factorize f(x) using the potential rational roots ±1, ±2, ±3, ±6, and ±10, we would get[tex](x+1)(x-1)(x+2)(x-2)(x+3)(x-3)(x+6)(x-6)(x+10)(x-10)[/tex]. This factorization can then be used to solve the equation.
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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 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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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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What is the nature of roots of the equation 4x² 4x 1 0?
The nature of roots of the equation 4x² -4x -19 = 0 is : The roots are real.
Given quadratic equation to us is 4x² -4x -19 = 0.
With respect to Standard form ax² +bx +c = 0 , Discriminant of the quadratic equation is given by b² - 4ac . Here's a table for Nature of Roots .
SI. NO. CONDITION NATURE OF ROOTS
1 Discriminant > 1 Real roots
2 Discriminant = 1 Equal roots
3 Discriminant < 1 Complex roots
Hence here , with respect to Standard form ,
• a = 4 , • b = -4 & •c = -19 .
⇒ Discriminant = b² - 4ac
⇒ Discriminant = (-4)² - 4 (4) (-19)
⇒ Discriminant = 16 + 304
⇒ Discriminant = 320
Hence since Discriminant is Greater than 0 , hence the nature of roots of the quadratic equation is real.
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