Correct Question:-
Solve the following differential equation using Laplace transformation.
[tex]\rm{x''(t)+x'(t)=2,\quad x(0)=3,\quad x'(0)=1}[/tex]
where [tex]\rm{x'(t)=\dfrac{dx}{dt},\quad x''(t)=\dfrac{d^2x}{dt^2}.}[/tex]
[tex]\quad[/tex]
Solution:-
Let,
[tex]\cal{L}\{\rm{x(t)}\}=X(s)[/tex]
[tex]\cal{L}^{\rm{-1}}\{\rm{X(s)}\}=x(t)[/tex]
Given,
[tex]\longrightarrow\rm{x''(t)+x'(t)=2}[/tex]
We take Laplace transformation of both sides of the equation.
[tex]\longrightarrow\cal{L}\{\rm{x''(t)+x'(t)}\}=\cal{L}\{\rm{2}\}[/tex]
[tex]\longrightarrow\cal{L}\{\rm{x''(t)}\}+\cal{L}\{\rm{x'(t)}\}=\cal{L}\{\rm{2}\}[/tex]
We have,
[tex]\cal{L}\{\rm{x''(t)}\}=\rm{s^2\,X(s)-s\,x(0)-x'(0)}[/tex]
[tex]\cal{L}\{\rm{x'(t)}\}=\rm{s\,X(s)-\,x(0)}[/tex]
[tex]\cal{L}\{\rm{2}\}=\rm{\dfrac{2}{s}}[/tex]
Then,
[tex]\small\text{$\longrightarrow\rm{\big[s^2\,X(s)-s\,x(0)-x'(0)\big]+\big[s\,X(s)-x(0)\big]=\dfrac{2}{s}}$}[/tex]
[tex]\longrightarrow\rm{s(s+1)\,X(s)-(s+1)\,x(0)-x'(0)=\dfrac{2}{s}}[/tex]
Given that [tex]\rm{x(0)=3}[/tex] and [tex]\rm{x'(0)=1.}[/tex] Then,
[tex]\longrightarrow\rm{s(s+1)\,X(s)-3(s+1)-1=\dfrac{2}{s}}[/tex]
[tex]\longrightarrow\rm{s(s+1)\,X(s)-3s-4=\dfrac{2}{s}}[/tex]
[tex]\longrightarrow\rm{s(s+1)\,X(s)=\dfrac{2}{s}+3s+4}[/tex]
[tex]\longrightarrow\rm{X(s)=\dfrac{2}{s^2(s+1)}+\dfrac{3s+4}{s(s+1)}}[/tex]
[tex]\longrightarrow\rm{X(s)=\dfrac{3s^2+4s+2}{s^2(s+1)}\quad\dots(1)}[/tex]
We will factorise this expression by partial fractions.
Assume,
[tex]\longrightarrow\rm{\dfrac{3s^2+4s+2}{s^2(s+1)}=\dfrac{As+B}{s^2}+\dfrac{C}{s+1}}[/tex]
[tex]\longrightarrow\rm{\dfrac{3s^2+4s+2}{s^2(s+1)}=\dfrac{(As+B)(s+1)+Cs^2}{s^2(s+1)}}[/tex]
[tex]\longrightarrow\rm{3s^2+4s+2=(As+B)(s+1)+Cs^2}[/tex]
[tex]\longrightarrow\rm{3s^2+4s+2=(A+C)s^2+(A+B)s+B}[/tex]
Equating corresponding coefficients,
[tex]\rm{A+C=3}[/tex]
[tex]\rm{A+B=4}[/tex]
[tex]\rm{B=2}[/tex]
Solving each equation we get,
[tex]\rm{A=2}[/tex]
[tex]\rm{B=2}[/tex]
[tex]\rm{C=1}[/tex]
Therefore,
[tex]\longrightarrow\rm{\dfrac{3s^2+4s+2}{s^2(s+1)}=\dfrac{2s+2}{s^2}+\dfrac{1}{s+1}}[/tex]
[tex]\longrightarrow\rm{\dfrac{3s^2+4s+2}{s^2(s+1)}=\dfrac{2}{s}+\dfrac{2}{s^2}+\dfrac{1}{s+1}}[/tex]
Then (1) becomes,
[tex]\longrightarrow\rm{X(s)=\dfrac{2}{s}+\dfrac{2}{s^2}+\dfrac{1}{s+1}}[/tex]
Now we will inverse Laplace transformation to obtain the solution.
[tex]\longrightarrow\cal{L}^{\rm{-1}}\{\rm{X(s)}\}=\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{2}{s}+\dfrac{2}{s^2}+\dfrac{1}{s+1}}\right\}[/tex]
[tex]\small\text{$\longrightarrow\rm{x(t)}=\rm{2}\,\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1}{s}}\right\}+\rm{2}\,\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1}{s^2}}\right\}+\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1}{s+1}}\right\}$}[/tex]
[tex]\small\text{$\longrightarrow\rm{x(t)}=\rm{2}\,\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1}{s}}\right\}+\rm{2}\,\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1!}{s^{1+1}}}\right\}+\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1}{s-(-1)}}\right\}$}[/tex]
We have,
[tex]\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1}{s}}\right\}=\rm{1}[/tex]
[tex]\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1!}{s^{1+1}}}\right\}=\rm{t}[/tex]
[tex]\cal{L}^{\rm{-1}}\left\{\rm{\dfrac{1}{s-(-1)}}\right\}=\rm{e^{-t}=\dfrac{1}{e^t}}[/tex]
Hence,
[tex]\longrightarrow\rm{\underline{\underline{x(t)=2+2t+\dfrac{1}{e^t}}}}[/tex]
This is the solution to our differential equation.
[tex]\quad[/tex]
Some results of Laplace Transformation:-
[tex]\boxed{\begin{array}{c|c}&\\\rm{x(t)}=\cal{L}^{\rm{-1}}\{\rm{X(s)}\}&\rm{X(s)}=\cal{L}\{\rm{x(t)}\}\\\\=============&==================\\\\\rm{\dfrac{d^nx}{dt^n}}&\rm{s^n\,X(s)}-\displaystyle\sum_{\rm{r=0}}^{\rm{n-1}}\rm{s^{n-r-1}\,\dfrac{d^rx}{dt^r}(t=0)}\\\\----------&--------------\\\\\rm{x'(t)}&\rm{s\,X(s)-x(0)}\\\\----------&--------------\\\\\rm{x''(t)}&\rm{s^2\,X(s)-s\,x(0)-x'(0)}\\\\----------&--------------\\\\\rm{a}&\rm{\dfrac{a}{s}}\\\\----------&--------------\\\\\rm{t^n,\ n}\in\mathbb{N}&\rm{\dfrac{n!}{s^{n+1}}}\\\\----------&--------------\\\\\rm{e^{at}}&\rm{\dfrac{1}{s-a}}\\&\end{array}}[/tex]
Construct a table and find the indicated limit. If h(x) = square root x + 2/x - 5, then, Complete the table below.
The limit of h(x) as x approaches infinity is ∞, and as x approaches 0, is undefined due to division by zero.
x h(x)
1 2 + 2
2 1.41 + 1
4 2 + 0.25
8 2.83 + 0.125
16 4 + 0.0625
To find the limit of h(x) as x approaches infinity, we see that h(x) = √(x) + 2/x - 5. As x increases without bounds, the square root of x and 2/x both approach infinity, but the value of -5 is constant. Therefore, the limit of h(x) as x approaches infinity is infinite.
To find the limit of h(x) as x approaches 0, we see that h(x) = √(x) + 2/x - 5. As x approaches 0, the value of the square root of x approaches 0, and the value of 2/x approaches infinity. However, the value of -5 is constant. Therefore, the limit of h(x) as x approaches 0 is -5 + ∞, which is undefined.
The limit of h(x) as x approaches 0 is undefined.
You could also say that the function is not defined at x = 0, meaning that the limit as x approaches 0 does not exist.
It's worth noting that the original function is not defined when x=0 as we are dividing by x, so it's impossible to calculate the limit of this function at that point.
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Select the reason that best supports Statement 8 in the given proof.
Answer:
C. Substitution
Step-by-step explanation:
Substitution is the final answer to an equation. So, the answer is C
Answer: C. Substitution
The length of the iland top i 35. 5 inche. The width of the iland top i 46. 75 inche. What i the area henry' rectangular kitchen iland top? Ue paper to how etimation or fraction multiplication to upport your anwer
The area henry' rectangular kitchen island top with length 35. 5 inches and width 46. 75 inches is 1659. 625 inch².
What is area of any shape ?The word "area" means an open area. A shape's area is calculated using its length and width. Length is one-dimensional and measured in units such as feet (ft), yards (yd), and inches (in). Area is defined as the total space occupied by a planar (2-D) surface or shape of an object. It is measured in units such as square feet (ft²) and square inches (in²)
The area of a rectangle is the space it occupies.The area of a rectangle is obtained by multiplying the length and width in equal units. So the formula is : Area of rectangle = length x width.Let " l " be length of rectangular island top and "w" be width of it.
We have , l = 35.5 inches , w = 46.75 inches
We have to calculate the area if henry' rectangular kitchen iland top.
let "A" be area of rectangular island top .
As discuss above, Required Area , A = l × w
=> A = 35.5 inches × 46.75 inches
=> A = 1659.625 inch²
Hence, required area is 1659.625 square inches.
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Tommy and zach are tarting out at the ame poition tommy run north at 5 mile per hour and zach tart to
run eat 2 hour later at the rate of 8 mile per hour how long until tommy and zach are 16 mile apart round to the nearet tenth if neceary
Tommy and Zach will be 16 miles apart after 2.90 hours.
Let the time taken for Tommy and Zach to run until they are 16 miles apart be t hours.
Tommy runs North at a speed of 5 miles per hour.
Therefore, the distance covered by Tommy in t hours is 5 multiplied by t hours:
=5t miles
Zach runs East at a speed of 8 miles per hour.
Zach started running 2 hours later so that he would run for (t - 2) hours.
Distance covered by Zach in (t - 2) hours running at a speed of 8 miles per hour :
=8(t - 2) miles
Since they are now 16 miles apart, Pythagoras's theorem, a²+ b² = c², is applied.
[8(t - 2)]²+ (5t)² = 16²
64(t - 2)² + 25t² = 256
64(t² + 4 - 4t) + 25t² = 256
64t² + 256 - 256t + 25t² = 256
89t² -256t + 256 - 256 = 0
89t² - 256t = 0
t( 89t - 256) = 0
(89t - 256) = 0
9t = 256
= 256÷9
t = 2.90 hours
Therefore, for 2.90 hours, Zach and Tommy will be 16 miles apart.
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How do you check the function is continuous or discontinuous?
A function is continuous if it is defined for all values of the independent variable and there are no breaks, jumps, or gaps in its graph. In other words, a function is continuous if you can draw its graph without lifting your pen from the paper.
To check whether a function is continuous at a particular point, you can use the following methods:
The two-part definition of continuity: A function is continuous at a point if the following two conditions are met:
The limit of the function as it approaches the point from the left and the right exists and is equal to the function's value at that point.
The function is defined at the point.
The Epsilon-Delta definition of continuity: A function is continuous at a point if, for every epsilon (a small positive number) there exists a delta (also a small positive number) such that, for all points x within delta units of the point in question, the function's value at x is within epsilon units of the function's value at the point.
The graph of the function: You can also check the continuity of a function by looking at its graph. If the graph is a continuous curve with no breaks or jumps, then the function is continuous. If the graph has breaks or jumps, then the function is discontinuous.
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The Epsilon-Delta definition of continuity states that a function is continuous at a point if there is an epsilon (a small positive number) and a delta (also a small positive number) such that the value of the function at x is within epsilon units of the value of the function at the point for all points x within delta units of the point in question.
The function's graph: Additionally, you can examine a function's graph to see whether it is continuous. The function is continuous if the graph displays a continuous curve without any interruptions or leaps. The function is discontinuous if the graph has breaks or jumps.
What is the relation of the volume of the pyramid its height and the area of its base?
The volume of the pyramid is one-third of the product of its height and the area of its base. That is volume of the pyramid is linearly proportional to its height and the area of its base.
A pyramid is a solid with a base and rectangle face on the edge of base meeting at a vertex. Generally we consider a right pyramid.
Consider a pyramid of base area A and height h, the volume of the pyramid is one-third of the product of its height and the area of its base, that is the volume formula of pyramid is
V = A × h/ 3
That is
V ∝ A and
V ∝ h
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Find the coordinates of point $p$ along the directed line segment $ab$ , from $a\left(8,\ 0\right)$ to $b\left(3,-2\right)$ , so that the ratio of $ap$ to $pb$ is $1$ to $4$.
The required coordinates of point P, following the given coordinate points, are (5.5, -1).
Let's assume that point P has coordinates (x, y). We want to find the coordinates of point P along the directed line segment AB, such that the ratio of AP to PB is 1 to 4.
The coordinates of point A are (8, 0), and the coordinates of point B are (3, -2).
According to the ratio given, we have:
AP/PB = 1/4
Now, we can use the midpoint formula to find the coordinates of point P. The midpoint formula states that the coordinates of the midpoint M between two points (x₁, y₁) and (x₂, y₂) are given by:
Midpoint M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Let's use the midpoint formula to find the coordinates of point P:
x = (8 + 3) / 2
= 5.5
y = (0 + (-2)) / 2
= -1
So, the coordinates of point P are (5.5, -1).
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What is the slope of the line 5x 3y 15?
-a/b=-5/3 is the slope of the line 5x 3y 15
for
ax+by=c
the slope is -a/b
the slope of paralell lines are the same
find slope
5x+3y=15
-a/b=-5/3
paralell as same slope
the slopes are:
L1: slope 3
L2: slope -3
L3: slope -1/3
So now multiply the slopes of each pair of lines:
L1 and L2: 3 * (-3) = - 9 => No, they are not perpendicular
L2 and L3: (-3) * (-1/3) = 1 => No, they are not perpendicular
L1 and L3: (3) * (-1/3) = -1 => Yes, they are penpendicular.
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How do you factor x3 y3?
The factor of x³y³ is x.x.x.y.y.y or x²xy²y.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
x³y³
This can be factorized as,
x.x.x.y.y.y
or
x²xy²y
Thus,
The factor is x.x.x.y.y.y or x²xy²y.
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Explain how to find the circumference of the circle with a radius of 4 inches
The circumference of the circle with a radius of 4 inches is 8π inches.
What is the circumference of a circle?The circumference is the perimeter of a circle or ellipse in geometry. That is, the circumference would be the circle's arc length if it were opened up and straightened out to a line segment. In general, the perimeter is the length of the curve around any closed figure.
Given that the radius of the circle is 4 inches. The circumference of the circle will be calculated by the formula,
Circumference = 2πr
Circumference = 2 x π x 4
Circumference = 8π inches
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The senior class is collecting orders for graduation, So far, 54% have ordered their yearbook, 58% have ordered their gown and 15% have ordered their gown but not their yearbook. Find that probability that a randomly selected senior...
1. has ordered both a gown and a yearbook.
2. has ordered a yearbook or a gown.
3. has ordered neither a yearbook nor a gown.
4. has ordered a yearbook but not a gown.
Answer:
1. has ordered both a gown and a yearbook: 54%
2. has ordered a yearbook or a gown: 58% + 54% = 112%
3. has ordered neither a yearbook nor a gown: 100% - (58% + 54%) = 88%
4. has ordered a yearbook but not a gown: 54% - 15% = 39%
Step-by-step explanation:
What is the mean of 2 and 12?
The required mean of 2 and 12 is 7.
What is mean?One kind of average is the mean.
It is calculated by dividing the total number of values in a list of values, such as measurements or numbers, by the total number of values in the list.
Add up each value in the set to determine the mean.
So, we have the values:
2 and 12
Mean formula: Sum of terms / Number of terms
Now, solve for the mean of the 2 values using the formula as follows:
Mean: Sum of terms / Number of terms
Mean: 14 / 2
Mean: 7
Therefore, the required mean of 2 and 12 is 7.
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Olivia is going on a 490 mile journey. So far, she has travelled 1% of her journey. How many miles has she travelled
Answer:
4.9 miles
Step-by-step explanation:
(.01)(490)=4.9
The Cowboys win another game! Let's goooo!!!
Answer:
cowboys fan:0 im guessing lol
Step-by-step explanation:
cowboys aren't bad so i'm not complaining:)
What is the slope of the line given by the equation 3x 4y =- 12?
The line produced by the equation 3x + 4y=-12 has a slope of -3/4.
To find the slope of a line represented by the equation 3x + 4y = -12, you need to have the equation in the form y = mx + b, where m is the slope and b is the y-intercept (the point at which the line crosses the y-axis).
To put the equation 3x + 4y = -12 in this form, you can start by isolating the y term on one side of the equation:
3x + 4y = -12
4y = -3x - 12
y = (-3/4)x - (12/4)
The slope of the line produced by the equation 3x + 4y=-12 is therefore -3/4.
The complete question is:-
What is the slope of the line given by the equation 3x +4y =- 12?
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Multiple choice math. Make sure you are correct tho!
Answer: The answer is option 1
Step-by-step explanation
The given equation is [tex](p^{2} r^{3} /pr^{4} )^{2}[/tex]
[tex](p^{2} r^{3} /pr^{4})^{2}[/tex] = [tex]p^{4} r^{6} /p^{2}r^{8}[/tex] (the given equation whole square is applied)
[tex]p^{4} r^{6} /p^{2} r^{8}[/tex] = [tex]p^{2} /r^{2}[/tex] (the p and r are canceled out )
Therefore the correct answer is [tex]p^{2} /r^{2}[/tex]
1/cosx-tanx=3 what is the value of sinx
The value of sinx is 1 - 3cosx
What is Trigonometry?
Trigonometry is a field of mathematics that explores the connections between triangle side lengths and angles. The topic arose in the Hellenistic civilization during the third century BC from geometric applications to astronomical research. Today, four varieties of trigonometry are used: core, plane, spherical, and analytic. The ratio of the sides and angles of a right triangle is the subject of core trigonometry.
Solution:
1/cosx - tanx = 3
we can write tanx as sinx/cosx
this is because we know that sinx = Perpendicular / Hypotenuse
and cosx = Base / Hypotenuse
so if we divide sinx by cosx we will find that it is equal to Perpendicular / Base which is nothing but tanx
1-sinx = 3cosx
sinx = 1 - 3cosx
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What is the solution of 2x² 5x 3 0?
The solution of the equation 2x^2 + 5x + 3 = 0 is x = -3 and x = -1/2. This can be found by using the Quadratic Formula, which states that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions are x = (-b +/- sqrt(b^2 - 4ac))/2a.
The solution of the equation 2x^2 + 5x + 3 = 0 can be found by using the Quadratic Formula. The Quadratic Formula states that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions are x = (-b +/- sqrt(b^2 - 4ac))/2a. By substituting the given coefficients a = 2, b = 5, and c = 3 into this equation, we can solve for x. First, we calculate the discriminant b^2 - 4ac, which in this case is 25 - 24 = 1. Because the discriminant is positive, the equation has two real solutions. Next, we use the Quadratic Formula to calculate the solutions. To calculate the first solution, we substitute the given coefficients into the formula and solve for x: x = (-5 +/- sqrt(1))/2 = (-5 +/- 1)/2 = -3 and -1/2.
Therefore, the solution of the equation 2x^2 + 5x + 3 = 0 is x = -3 and x = -1/2.
Discriminant: b^2 - 4ac = 25 - 24 = 1
Solution 1: x = (-5 +/- sqrt(1))/2 = (-5 +/- 1)/2 = -3
Solution 2: x = (-5 +/- sqrt(1))/2 = (-5 +/- 1)/2 = -1/2
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determine the 2.5th percentile of the normal distribution with a mean of 72.5 and a standard deviation of 15 as a decimal rounded to the nearest tenth.
Answer:
43.1
Step-by-step explanation:
2.5 % = .025 look for this value on the z-score table to find this corresponds to z = - 1.96 Standard Deviations
72.5 - 1.96 * 15 = 43.1
Marie buys a kite priced at $27. If the sales tax is 10%, how much tax
will Marie pay?
Answer: $2.70
Step-by-step explanation:
10% = 0.1
Take 27 times 0.1 = $2.70
see this and pls give me the answers pls
can you give a more clearer image if possible, please?
The boxplot displays the arm spans for 44 students.
Which of the following is not a true statement?
O There are no outliers in this distribution.
O The shape of the boxplot is fairly symmetric.
Arm Spans of Students
The range of the distribution is around IS 60 cm.
O The center of the distribution is around 180 cm.
140 150 160 170 180 190 200 210
Arm Span (cm)
L
The statement that is true is there are no outliers in this distribution.
What is distribution ?
Selling and providing goods and services to customers is the process of distribution. The phrase is used to refer to marketing channels that are employed to contact clients in various contexts and geographical locations.
There are four different categories of distribution channels: direct selling, selling through middlemen, dual distribution, and reverse logistics networks.
The delivery of goods from manufacturers to end users is ensured by a network of distributors or intermediaries known as a distribution channel. It is also in charge of sending payments made by clients for purchases made by producers.
In this scenario, the noticeable outliers in the data sets are 200 and 210. Thus, a statement which is not true about the data provided in the box plot above, is that there are no outliers in this distribution
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What is the arithmetic mean of the data set 4 5 0 10 8 and 2?
Arithmetic mean of the data set 4, 5, 0, 10, 8 and 3 is '5'
As per the question the given data set is :
4 , 5 , 0 , 10 , 8 and 3
Number of observations in given data set are '6'
We have to determine the value of arithmetic mean of the data
Arithmetic Mean is simply the mean or average for a set of data or a collection of numbers.
Arithmetic mean is defined as the ratio of sum of all observations in given data set to the total number of values.
Arithmetic mean ( AM ) formula is
AM = (sum of all observations in given data set) ÷ (total number of observations)
Sum of all observations in given data set :
= 4 + 5 + 0 + 10 + 8 + 3
= 30
Arithmetic mean :
[tex]=\frac{30}{6}[/tex]
= 5
Therefore arithmetic mean of the given data set 4, 5, 0, 10, 8 and 3 is 5.
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How do you know if a figure is similar or congruent?
Answer:
Congruent means the same shape but perhaps moved or rotated or reflected
Similar means a shape that has been enlarged by a scale factor
Step-by-step explanation:
One card is drawn (hkv-fhyp-che) from a well-shuffled deck of 52 cards. Calculate the probability that the card will
(i) be an ace,
(ii) not be an ace.
The probability of getting an ace from a well-shuffled deck of 52 cards is 1/13 and the probability of not getting an ace is 12/13
We can calculate the probability by the following method:
Total number of cards in a pack of cards= 52
Number of ace cards in a pack of cards= 4
Probability of an event= Number of possibilities of that event / Total number of possibilities
(i) If drawing an ace card is considered an event, suppose E, then:
P(E)= Number of ace cards/ Total number of cards
= 4/52
(ii) Probability of any event other than E= 1- Probability of the event E=
If the drawing of a non-ace card is considered an event, suppose E', then
P(E')= 1- the probability of drawing an ace card
= 1- P(E)
= 1-1/13
= (13-1)/ 13
= 12/13
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How do you find the value of X and Y in a given figure?
The value of angle x is 70° and angle y is 60° in the given figure with exterior angle = 120°
From the given figure, we know that the Exterior angle = 120°
To find the values of x and y we will follow the following steps.
We know that the sum of interior opposite angles is equal to the exterior angle of the triangle.
The exterior angle of the triangle=120°
The interior angles are x and 50°
50° + x = 120°
x = 120°- 50°
x = 70°
Also, we know that the sum of the y and the exterior angle is 180° because they are forming a linear pair.
Hence we have,
y + 120° = 180°
y = 180° - 120°
y = 60°
Therefore, the value of angle x is 70° and angle y is 60°.
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What is the nature of roots of the quadratic equation 9x2 6x 2 0?
The nature of roots of the given quadratic equation is "real and unequal".
Nature of Roots:
Root type simply means the category to which the root belongs. Roots can be imaginary, real, unequal, or equal. If the discriminant is negative, the root is imaginary.
Depending on the value of the discriminant, we discuss the following cases for the nature of the root:
Case 1: D = 0
If the discriminant is zero (b2 – 4ac = 0), a, b, and c are real, and a≠0, the root of the quadratic equation is ax2 + bx + c = 0 is Real and equal. In this case the root is x = -b/2a. A graph of the equation is tangent to the x-axis at a point.
Case 2: D > 0
If the discriminant is greater than zero (b2 – 4ac > 0), then a, b, and c are real and a≠0, then the root of the quadratic equation ax2 + bx + c = 0 is real and not equal. The equation graph touches the x-axis at two different points.
Case 3: D < 0
If the discriminant is less than zero (b2 – 4ac < 0), a, b, and c are real, and a≠0, then the root of the quadratic equation ax2 + bx + c = 0 is Imaginary and not equal. Roots exist in conjugate pairs. The equation graph does not touch the x-axis.
Case 4: Perfect Square with D > 0
If D > 0 and Perfect Square, the roots of the quadratic equation are real, unequal, and rational.
Case 5: D > 0 and Not Perfect Square
If D > 0 and not perfect square, the roots of the quadratic equation are real, unequal, and irrational.
According to the Question:
The given quadratic equation:
9x²-6x-2 = 0
Here, a = 9, b = - 6 and c = -2
To Find: the nature of roots of the given quadratic equation.
∴ Discriminant, D = b²-4ac
= (-6)² - 4×9×(-2)
= 36 -36(-2)
= 36 + 72
=108
Since,
D = 108 > 0, the roots are real and unequal.
Hence, the nature of roots of the given quadratic equation is "real and unequal".
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Cuánto es?
35
— –1
10
Answer:
145
Step-by-step explanation:
Cuánto es means what is the answer
35 - (-110)
Two negative signs equal a positive sign
35 + 110 = 145
How do you find the height of a box when given the volume and base?
In order to determine the height of a box we can use the formula h = V/b or if the given box is a square box or a cube then we can calculate its height by finding the cube root of its volume.
In the formula , V = volume and b = base
Volume is calculated in three dimension , it is the measure of the space occupied by an object . Any given solids can take the form of a cube, cuboid, cone, cylinder, or sphere.
There are various shapes in geometry and all of them have a specific formula to calculate the volume.
The SI unit of volume for a cube is m³.
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The probability that a person has immunity to a particular disease is 0.8. Find the mean number who have immunity in samples of size 17
The mean number who have immunity in samples of size 17 is 13.6
What is a Mean number?Sample Size =17
Immunity to a Particular disease=0.8
=17*0.8
=13.6
The average of the numbers is the mean. Calculating is simple: add up all the numbers, then divide by the total amount of numbers. To put it another way, it is the total divided by the number. In mathematics and statistics, the concept of mean is crucial. The most typical or average value among a group of numbers is called the mean.It is a statistical measure of a probability distribution's central tendency along the median and mode. It also goes by the name "anticipated value.It is a statistical idea with significant financial implications. The idea is applied in a number of financial areas, such as business appraisal and portfolio management, although not exclusively.To learn more about Mean refer to:
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