The nature of the roots of a quadratic equation is known by a value known as the discriminant, which is calculated by seeing the equation and comparing it with the standard equation which is given by -
ax^2 + bx +c , where a , b , c are constants .
The root of the equation is given by -
x= [(-b )± (b^2 - 4ac)] / 2a
Here, the term (b^2 - 4ac) is called the discriminant of the quadratic equation .
This discriminant is the defining factor of the nature of the roots.
Now, here arise three cases-
(i) If (b^2 - 4ac) > 0, the roots are real and distinct .
(ii) If (b^2 - 4ac) = 0 , the roots are real and equal.
(iii) If (b^2 - 4ac) < 0 , the roots are not real and they are imaginary i.e. they are complex in nature and of the form a+ib.
For example , we have an equation x^2 + 2x + 1.
Here, a =1, b=2 and c=1.
Putting these values in the discriminant , we get -
b^2 - 4ac = 4-4(1)(1) = 0.
So, the roots of the equation are real and equal i.e. x=2,2.
Hence, it is the discriminant which determines the nature of roots.
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The which raito is equivalent to 8/6?
Answer:
Hence, three equivalent ratios of 8: 6 are 16: 12, 24: 18, and 32: 24.
Step-by-step explanation:
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The general solution of the given non homogeneous
equation is
y(t)=c₁[tex]e^{-t}[/tex] +c₂[tex]e^{t}[/tex] - t
A non-homogeneous system of equations is a system in which the vector of constants on the right-hand side of the equal sign is non-zero.
Given:-General solution for the equation: [tex]y^{"} - y^{'} = t[/tex], [tex]y_{p} (t) = - t[/tex]
Explanation:-Consider the equation [tex]y^{"} - y^{'} = t[/tex] . The corresponding homogeneous equation has the associated auxiliary equation [tex]r^{2} -1 =0[/tex] which has [tex]r^{} =-1[/tex] and [tex]r^{} =1[/tex] as solutions.
The solution of an n-order differential equation depends on an algebraic equation of degree n called the characteristic equation (or auxiliary equation) in mathematics.
Hence the general solution for a homogeneous equation is
y(t) = [tex]-t[/tex]+[tex]c[/tex]₁[tex]e^{t}[/tex]+[tex]c[/tex]₂[tex]e^{-t}[/tex]
Combining the general solution [tex]y(t)=c[/tex]₁[tex]e^{t}+c[/tex]₂ [tex]e^{t}[/tex] with the particular solution of [tex]y_{p} (t) =-t[/tex] of [tex]y'-y"=t[/tex], we find that the general solution to[tex]y"-y'=t[/tex] is [tex]y(t)=c[/tex]₁[tex]e^{-t}+c[/tex]₂[tex]e^{t}[/tex]
The general solution of the given non homogeneous equation is
y(t)=c₁[tex]e^{-t}[/tex] +c₂[tex]e^{t}[/tex] - t
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What is degree 3 equation called?
Answer:
An equation with a degree of 3 is called a cubic function.
Step-by-step explanation:
Functions are named by their degree.
Examples:
[tex]y = ax^2 + bx + c[/tex] is named a quadratic function because it has a degree of 2.
Similarly, [tex]y = ax^3 + bx^2 +cx + d[/tex] is named a cubic function because it has a degree of 3.
What is 3/4 in lowest terms as a fraction?
3/4 is the lowest form of the given fraction.
What is lowest form of fraction?if the highest common factor of numerator and denominator if equals to 1. then that fraction can't be reduced further and is said to be in the lowest form.
so suppose we are having any fraction of the form [tex]\frac{a}{b}[/tex]
the lowest form of the given fraction is given by [tex]\frac{x}{y}[/tex]
where x = [tex]\frac{a}{gcd(a,b)}[/tex] , y = [tex]\frac{b}{gcd(a,b)}[/tex]
where gcd is greatest common divisor of a , b
we are given fraction = 3/4
=> a = 3 and b = 4
gcd(3,4) = 1
so x = 3/(gcd(3,4)) = 3
y= 4/(gcd(3,4)) = 4
so the lowest form of 3/4 = [tex]\frac{x}{y}[/tex]
and x=3 and y=4
and hence lowest form = 3/4
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The salaries of the employees at four companies are summarized below. Answer the questions about them
Company A: The range of salaries is $60,000 and the mean salary is $41,000.
Company B: The range of salaries is $53,000 and the mean salary is $39,000.
Company C: The range of salaries is $54,000 and the mean salary is $47,000.
Company D: The range of salaries is $58,000 and the mean salary is $38,000.
(a) Based on the information above, which company's salaries have the most variability?
a: Company A
b:Company B
c:Company C
d:Company D
(b) Based on the information
above, which company has the lowest salaries on average?
a:Company B
b:Company A
c:Company C
d:Company D
Part (a)
Company A has the most variability because it has the m greatest range.
Part (b)
Company B has the lowest salaries on average since they have the smallest mean salary.
i have no clue what i’m supposed to do
How do you know if the graph has a hole or vertical asymptote?
A hole in a graph appears as a break in the graph line, while a vertical asymptote appears as a line segment that the graph approaches but never crosses.
A hole in a graph appears as a break in the graph line; it's the point where the graph does not exist. Typically, the x-value of this point is the x-coordinate of the hole. To find out if there is a hole, you need to see if the graph is continuous and check to see if there is a break in the graph line.A vertical asymptote appears as a line segment that the graph approaches but never crosses. This occurs when the function value approaches positive or negative infinity. To find out if there is a vertical asymptote, you need to look at the function and see if it is undefined at any particular x-value. If it is, then there is a vertical asymptote.
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There are 52800 fans at a football match between Rovers and City. 37% of the fans support Rovers. How many fans at the match support City?
Total Fans = 52800
Total fans supporting Rovers = 37%
Answer: City fans = 33,264 fans
because City fans percentage = 100% - 37% = 63%
Multiplier = 0.63 (63/100)
So City fans = 0.63 x 52800 = 33,264
Hope this helps!
Answer:
Step-by-step explanation:
To find out how many fans at the match support City, we first need to find out how many fans support Rovers. We can do this by multiplying the total number of fans by the percentage of fans who support Rovers:
Total fans supporting Rovers = 52800 fans * 37% = 19296 fans
Since the total number of fans at the match is 52800, and 19296 of those fans support Rovers, then the number of fans who support City is:
Total fans supporting City = 52800 fans - 19296 fans = 33504 fans
Therefore, there are 33504 fans at the match who support City.
How do you write 2x 3y 12 in slope intercept form?
The required equation in the slope-intercept form is y = 2/3x - 4.
What is slope intercept form?Y = MX + b, where m denotes the slope and b the y-intercept, is how the equation of the line is expressed in the slope-intercept form.
We can see that the slope of the line in our equation, y = 6x + 2, is 6.
So, we have the equation:
2x − 3y = 12
To represent the equation in the slope-intercept form:
First, remove 2x from both sides of the equation to move the independent variable (x) to the right side of the equation.
2x − 3y − 2x = 12 − 2x
− 3y = 12 − 2x
The dependent variable is the only component of the left side of the equation, as seen in the generic equation of a line in slope intercept form, whereas the independent variable and constant are both present on the right side of the equation.
The result of multiplying by 1 and dividing by 3 into both sides of the equation is:
-1 * -3y/3 = -1 * (12 - 2x)/3
y = 2/3x - 12/3
y = 2/3x - 4
Therefore, the required equation in the slope-intercept form is y = 2/3x - 4.
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LO. Gabe has 60 coins consisting of quarters and dimes.
The combined value of the coins is $9.45.
How many
quarters and dimes does Gabe have?
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Answer:
adult ticket $16
child ticket $12
Step-by-step explanation:
let x = price of adult tix
y = price of child tix
1) 3x + 4y = $96
2) 2x + 3y = $68
Subtract equation 2 from equation 1 to get:
3) x + y = 28
x = 28 - y (now substitute this into equation 1)
3(28-y) + 4y = 96
84 - 3y + 4y = 96
y = 96 - 84 = 12 (price of child tickets)
Now that you know the value of y, plug it into equation 3:
x + 12 = 28
x = 28 - 12 = 16 (price of adult tickets)
How do you tell if a system has no solution or infinitely many?
When a system has no solution it is called inconsistent equation
When a system has infinitely many solution it is called consistent and dependent equations
How to tell if a system is are inconsistent equationsA system is considered to be inconsistent if there is no solution. There is no solution because the line graphs are parallel and not intersecting.
x + 5y = 4 and x + 5y = −2
subtracting gives
0 = -6 no solution
this is an instance
How to tell if a system is consistent and dependent equationsA consistent system is dependent if there are an endless number of possible solutions. The two equations represent the same line when you graph them. An instance of the situation is
3x + y = 4
−6x − 2y = −8
subtracting gives
9x + 3y = 12 - none eliminates, meaning infinite many solutions
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The figure is made of a rectangle and a half of a circle find the perimeter of the figure to the nearest tenth . Use 3.14 for pi . Also Please show ur work Bc I have too !!
The perimeter of the figure as shown in the diagram is 60.84 m.
What is perimeter?Perimeter is the total distance around an object.
To calculate the perimeter of the figure, we use the formula below.
Formula:
P = 2L+W+Wπ/2.............. Equation 1Where:
P = Perimeter of the figureL = Length of the figureW = Width of the figureFrom the question,
Given:
L = 15 mW = 12 mπ = 3.14Substitute these values into equation 1
P = (15×2)+12+(12×3.14/2)P = 30+12+18.84P = 60.84 mHence, the perimeter of the figure is 60.84 m.
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The following table shows the values of y for different values of x:
x
y
0
0
1
0
2
4
Which statement best explains whether the table represents a linear function or a nonlinear function?
Group of answer choices
It represents a linear function because its points are not on a straight line.
It represents a nonlinear function because its points are not on a straight line.
It represents a linear function because its points are on a straight line.
It represents a nonlinear function because its points are on a straight line.
Option B. It represents a nonlinear function because its points are not on a straight line.
What is linear and non-linear functions?
When plotted on a graph, a linear function produces a straight line. Therefore, any non-linear functions are those that are functions. The quickest way to tell if a function is linear or non-linear is probably by graphing it, but we can also tell if a function is linear by looking at its input/output table or equation.
If A(0, 0), B(1, 0) and C(2, 4).
We know that,
If point A,B and C are on a straight line then the slope of AB must be equal to the slope of AC
The formula to calculate the slope between two points is equal to
[tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex]
Find the slope AB
A(0, 0) and B(1, 0)
[tex]m = \frac{0-0}{1-0} = 0[/tex]
Find the slope AC
A(0, 0) and C(2, 4)
[tex]m = \frac{2-0}{4-0} = \frac{1}{2}[/tex]
Slope of AB ≠ Slope of AC
Points A, B and C are not on a straight line.
Hence, it represents a nonlinear function because its points are not on a straight line.
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What is the elevation of point a? point a has an elevation that is greater than 3350 feet. Point a has an elevation that is less than 3350 feet. Point a has an elevation of 3350 feet.
The elevation of point A is greater than 3350 feet.
The lines of equal elevation are known as contour lines. In relation to the mean sea level, it reflects the height of any specific site. The crucial elements on a topographic map are those that never cross each other.
Locations A and B are shown on the provided topographic map. By reading the contour and the contour interval, which is shown to be 50 feet, it is clear that Location A is higher up than Location B.
Location A is at a higher contour line, with respect to the contour 3250 feet, mentioned on the left side of location A.
Thus, the height of location A is slightly more than 3350 feet above the mean sea level.
(Refer to the image attached for the topographic map)
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Context lines are those lines that have the same elevation. It reflects the height of any particular site in relation to the average sea level. A topographic map's most important features are those that never cross one another.
The topographic map is displayed with the locations A and B. It is evident that Location A is higher up than Location B by reading the contour and the contour interval, which is indicated to be 50 feet.
In comparison to the contour line of 3250 feet listed on the left side of location A, position A is at a higher contour line.
As a result, position A is higher above the norm by about 3350 feet.
There are 2 squares and 4 triangles. What is the simplest ratio of squares to triangles?
Answer:
squares : triangles
1:2 or 1/2
Step-by-step explanation:
⭐What is a ratio?
A ratio is a comparison of two units.In this problem, the two units are squares and triangles.⭐ How can we write a ratio?
You can write a ratio as a fraction (e.g. 2/2)You can write a ratio with a colon (e.g. 2:2)For this problem, we can write the ratio of squares to triangles as:
[tex]\frac{2}{4}[/tex], or [tex]2:4[/tex].
But, we can further simplify this ratio.
To simplify the ratio, divide the numerator and denominator or both sides of the colon by the number they are both divisible by.
2 is divisible by 2, and 4 is divisible by 2:
[tex]\frac{2}{4}[/tex]
[tex]\frac{2 /2}{4/2}[/tex]
Simplified is: [tex]\frac{1}{4}[/tex]
[tex]2:4[/tex]
[tex]\frac{2}{2} : \frac{4}{2}[/tex]
Simplified is: [tex]1:4[/tex]
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Which statement describes how to determine if a relation given in a table is a function?
If none of the output values are repeated, the relation is a function.
If none of the input values are repeated, the relation is a function.
If any of the input values are equal to the output values, the relation is a function.
If all the output values are paired with the same input value, the relation is a function.
Answer:
“If none of the input values are repeated, the relation is a function” because one input must have one output.
How do you graph an inequality example?
The graph of an inequality is shown below -
What is an inequality equation?The graph of a two-variable inequality is the set of points that describe all solutions to the inequality. The coordinate plane is split in half by a boundary line created by a linear inequality, with one-half representing the solutions to the inequality.
It's a line with a shaded side on one side to show which x-y pairings are the inequality's solutions.
Example -
Let us take a equation, 4x + 8y ≤ -24
Now, solve the above equation in the form of slope-intercept.
4x + 8y ≤ -24
or, 8y ≤ - 4x - 24
or, y ≤ -(1/2)x - 3
Now, the graph becomes according to the slope-intercept form is:
Here we noticed that,
This side of the inequality is shaded below (instead of above) because y is lower than (or equal to) the other side.
Because working with a "or equal to" inequality, draw a solid line rather than a dashed one. Points on the solid line denote locations where the inequality has been solved.
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What is the sum of 5/14+24 1/2 ?
Enter your answer as a mixed number in simplest form by filling in the boxes.
Answer:
24 6/7
Step-by-step explanation:
5/14+24 1/2
24 1/2 = 49/2
5/14 + 49/2
The common factor is 14
5/ 14 + 343 / 14 = 348 /14 = 174 / 7
174/ 7 = 24 6/7
Step-by-step explanation:
1. Convert the mixed number 24 1/2 to an improper fraction by multiplying the whole number (24) by the denominator of the fraction (2) and adding the numerator (1). This gives us an improper fraction of 49/2.
2. Find a common denominator for the fractions 5/14 and 49/2. The least common multiple of 14 and 2 is 14, so we can rewrite the fractions as 7/7 and 49/14, respectively.
3. Add the numerators and keep the denominator the same: 7/7 + 49/14 = (7+49)/7 = 56/7
4. Simplify the fraction 56/7 by dividing both the numerator and the denominator by their greatest common factor, which is 7. This gives us a simplified fraction of 8/1, or 8.
5. The sum of 5/14+24 1/2 is equal to 8.
Tara picks out 4 paint colors (beige, blue, green, and gray) to paint 4 rooms in her house (living room, dining room, main bedroom, and kitchen). The probability that Tara first paints her bedroom green is .
The probability that Tara first paints her bedroom green is 1/16.
What is the probability that Tara first paints her bedroom green?Probability is the occurence of likely events. It is the area of mathematics that deals with numerical estimates of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1.
In this case, Tara picks out 4 paint colors (beige, blue, green, and gray) to paint 4 rooms in her house (living room, dining room, main bedroom, and kitchen).
The probability of painting bedroom = 1/4
The probability of picking green = 1/4
Therefore, the probability that Tara first paints her bedroom green is:
P(bedroom) × P(green)
= 1/4 × 1/4
= 1/16
The probability is 1/16.
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The number of y of Salmonella cells on an egg after t minutes is y=a(4.7)^t/45 , where a is the initial number of cells. By what percent does the number of Salmonella cells increase each minute?
3.5% percent does the number of Salmonella cells increase each minutes when y = a(4.7[tex])^{t/45}[/tex].
Given that,
The number of y of Salmonella cells on an egg after t minutes is
y = a(4.7[tex])^{t/45}[/tex]
Here,
y is the number of salmonella cells.
a is the initial number of cells.
t is the time in minutes.
We have to find what percent does the number of Salmonella cells increase each minutes.
We know that,
Take the original function,
y = a(4.7[tex])^{t/45}[/tex]
Take the power of proportionality
y = [a(4.7[tex])^{1/45}]^{t}[/tex]
Evaluating the power
y = [a(4.7[tex])^{1/45}]^{t}[/tex]
y = a(1.035[tex])^{t}[/tex]
y = a(1-0.035[tex])^{t}[/tex]
Therefore, 3.5% percent does the number of Salmonella cells increase each minutes when y = a(4.7[tex])^{t/45}[/tex].
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Fill in the missing number. 70% of = 7
whats 2+2+2+4+4+2+4 and writting form
2+2+2+4+4+2+4 = 20 and It's written in the addition form.
What is addition ?
One of the four fundamental mathematical operations, along with addition, subtraction, multiplication, and division, is addition. This operator is used to combine two or more objects or numbers.
Given, 2+2+2+4+4+2+4
The result can be found by applying addition,
So, 2+2+2+4+4+2+4 = 20
We can also find the sum of the above numbers by applying multiplication,
We've, 2+2+2+4+4+2+4
We will rearrange it in such a way that all the same numbers come together,
So, 2+2+2+2+4+4+4
Now, we can see that the number 2 is occurring 4 times and the number 4 is occurring 3 times,
So, It can be written as,
(2 × 4) + (4 × 3)
= 8 + 12
= 20.
Hence, We get the same answer in both the cases.
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A recipe for cinnamon rolls calls for 3 cups of flour for every 1/2 cup of water. Using the same recipe, how much water will you need for 5 cups of flour?
Answer: 3/10 cup of water
Step-by-step explanation:
Take 3 divided by 5 = 0.6
So, we know to get from 3 to 5 we need to times 0.6
So, we take 1/2 times 0.6
1/2 = 0.5
0.5 times 0.6 = 0.3 cup of water
0.3 = 3/10
So, the answer is 3/10 cup of water
Does 6 only have 2 factors?
Answer:
no
Step-by-step explanation:
the factors of 6 are
1, 2, 3, 6
that is 6 has 4 factors
Select all the true statements about this graph.
A. The graph is nonlinear
B. The function increases at the same rate
C. The rate decreases after x = 2
D. The graph is a function
E. The graph is increasing in two intervals
(Picture of graph included.)
Answer:
A. The graph is nonlinear.
D. The graph is a function
E. The graph is increasing in two intervals.
Step-by-step explanation:
We will review each answer choice to decide on our answer to this question.
✓ A. The graph is nonlinear.
Since the graph has a change in it's rate of increase, it's nonlinear.
✗ B. The function increases at the same rate.
As stated below, the rate increases after x = 2 so the function is not increasing at the same rate.
✗ C. The rate decreases after x = 2.
No, the rate increases after x = 2.
✓ D. The graph is a function.
Yes, there is only one output for every input, so this is a function.
✓ E. The graph is increasing in two intervals.
The graph is only increasing in two intervals. These intervals change at x = 2.
a suburban specialty restaurant has developed a single drive-thru window. customers order, pay, and pick up their food at the same window. arrivals follow a poisson distribution while service times follow an exponential distribution. if the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes, what is the average number of customers waiting in line behind the person being served? a. 3.00 b. 0.5 c. 2.25 d. 30.0 e. 0.75
Using the Poisson distribution and the exponential distribution, on average, there are (C) 2.25 consumers in line behind the person serving.
What is exponential distribution?The Poisson point process, or a process in which events occur continuously and independently at a constant average rate, uses an exponential distribution to represent the probability distribution of the time between occurrences.
Statistics and probability theory both make use of this distribution.
So, we know that:
6 arrivals on average are made per hour (Poisson distribution).
It provides two services every fifteen minutes (exponential distribution).
Waiting line behind the person being served:
(15 / 6 * 2) + 1
(15 / 12) + 1
1.25 + 1
2.25
Therefore, using the Poisson distribution and the exponential distribution, on average, there are (C) 2.25 consumers in line behind the person serving.
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Why is it called an asymptote?
An asymptote function is called such because it is a line that a curve approaches but never touches, or "asymptotically" approaches.
An asymptote is a line that a curve approaches but never actually touches. It is called an asymptote because the word comes from the Greek asymptōtos, which means "not falling together." Asymptotes are used in mathematics to describe curves that approach a particular line without ever intersecting it. For example, in a graph of a function, an asymptote is a line that the graph of the function approaches, but never intersects with, as it approaches infinity. Asymptotes are also used to describe straight lines in a plane that approach each other at a certain angle, but never meet. In both cases, the idea is that the line or curve approaches the asymptote, but never actually touches it, thus the name asymptaote.
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How do you know if a function has a hole?
A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is typically represented as y = f. (x).
A function can have a "hole" if it is not defined at a certain input value. For example, consider the following function:
f(x) = 1/x
This function has a hole at x=0 because it is not defined at that value. If you try to evaluate the function at x=0, you will get an error or an undefined result.
To determine whether a function has a hole, you can consider the domain of the function, which is the set of all input values for which the function is defined. If there are any values in the domain for which the function is not defined, then the function has a hole at those values.
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What is 1/2 of a square?
One-half squared is one-fourth. In other words , 1/ 4 is the square is 1/2 .
What does Squaring mean?In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9.
Squaring a number is simply multiplying the value by itself. When you take the square root of a number, you are figuring out what number had to be squared to get the original value.
The square of a number (let x) is given by the formula x².
According to the Question ,
Square of 1/2 is given by = (1/2)²
Since , the square of a number (let x) is given by the formula x².
(1/2)²
= 1/(2)²
= 1/ 4
= 0.25
Hence, One-half squared is one-fourth. In other words , 1/ 4 is the square is 1/2 .
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Is y 3x 1 a direct variation?
The expression y-3x-1 is not a direct variation as it does not satisfy the criteria.
What is direct variation?It is a mathematical relationship between two variables shown by the form of an equation that says one variable is equal to the other variables that is multiplied by a factor.
Such as y =4x is an equation which express the direct variation between x and y as y is equal to four times of x
How can we identify the direct variation?The formula for direct variation is given by the equation y = kx.
x and y are variables, k is the constant or factor.
Suppose we examine a number of linear equations and check the ratio of the variables such as x and y ratio. If we find a constant ratio, then it is termed as direct variation.
In the given expression, y-3x-1 that could not satisfy the criteria of direct variation because there is a constant -1 in the expression. And while writing in an equation form, we get y-3x-1 =0
we never get constant ratio from x and y variables.
hence, this is not a direct variation.
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