TextNut 2.24

TextNut 2.24 强大的富文本编辑器. TextNut 是一款mac上专业强大的markdown编辑器,100%兼容CommonMark规范. Macbook苹果电脑文本编辑器 TextNut 2.24 for mac版. MarkDown 文本编辑器 它是独特的富文本编辑方式,写作时可以除掉markdown中标示字符,同时又可以随时切换回原.

  1. Text 22422
  2. Text 224444
  3. Textnut 2.24 Meters

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Two solutions were found :

  1. n = 6
  2. n = -4

Step by step solution :

Step 1 :

Trying to factor by splitting the middle term

1.1 Factoring n2-2n-24
The first term is, n2 its coefficient is 1.
The middle term is, -2n its coefficient is -2.
The last term, 'the constant', is -24
Step-1 : Multiply the coefficient of the first term by the constant 1-24 = -24
Step-2 : Find two factors of -24 whose sum equals the coefficient of the middle term, which is -2.

-24 + 1 = -23
-12 + 2 = -10
-8 + 3 = -5
-6 + 4 = -2 That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -6 and 4
n2 - 6n + 4n - 24
Step-4 : Add up the first 2 terms, pulling out like factors :
n • (n-6)
Add up the last 2 terms, pulling out common factors :
4 • (n-6)
Step-5 : Add up the four terms of step 4 :
(n+4) • (n-6)
Which is the desired factorization

Equation at the end of step 1 :

Step 2 :

Theory - Roots of a product :

2.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

2.2 Solve : n+4 = 0
Subtract 4 from both sides of the equation :
n = -4

Solving a Single Variable Equation :

2.3 Solve : n-6 = 0
Add 6 to both sides of the equation :
n = 6

Supplement : Solving Quadratic Equation Directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

3.1 Find the Vertex of y = n2-2n-24
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting 'y' because the coefficient of the first term, 1 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,An2+Bn+C,the n -coordinate of the vertex is given by -B/(2A) . In our case the n coordinate is 1.0000
Plugging into the parabola formula 1.0000 for n we can calculate the y -coordinate :
y = 1.0 * 1.00 * 1.00 - 2.0 * 1.00 - 24.0
or y = -25.000

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Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = n2-2n-24
Axis of Symmetry (dashed) {n}={ 1.00}
Vertex at {n,y} = { 1.00,-25.00}
n -Intercepts (Roots) :
Root 1 at {n,y} = {-4.00, 0.00}
Root 2 at {n,y} = { 6.00, 0.00}

Solve Quadratic Equation by Completing The Square

3.2 Solving n2-2n-24 = 0 by Completing The Square .
Add 24 to both side of the equation :
n2-2n = 24
Now the clever bit: Take the coefficient of n , which is 2 , divide by two, giving 1 , and finally square it giving 1
Add 1 to both sides of the equation :
On the right hand side we have :
24 + 1 or, (24/1)+(1/1)
The common denominator of the two fractions is 1 Adding (24/1)+(1/1) gives 25/1
So adding to both sides we finally get :
n2-2n+1 = 25
Adding 1 has completed the left hand side into a perfect square :
n2-2n+1 =
(n-1) (n-1) =
(n-1)2
Things which are equal to the same thing are also equal to one another. Since
n2-2n+1 = 25 and
n2-2n+1 = (n-1)2
then, according to the law of transitivity,
(n-1)2 = 25
We'll refer to this Equation as Eq. #3.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(n-1)2 is
(n-1)2/2 =
(n-1)1 =
n-1

Now, applying the Square Root Principle to Eq. #3.2.1 we get:
n-1 = 25
Add 1 to both sides to obtain:
n = 1 + √ 25
Since a square root has two values, one positive and the other negative
n2 - 2n - 24 = 0
has two solutions:
n = 1 + √ 25
or
n = 1 - √ 25

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Solve Quadratic Equation using the Quadratic Formula

3.3 Solving n2-2n-24 = 0 by the Quadratic Formula .
According to the Quadratic Formula, n , the solution for An2+Bn+C = 0 , where A, B and C are numbers, often called coefficients, is given by :

- B ± √ B2-4AC
n = ————————
2A
In our case, A = 1
B = -2
C = -24
Accordingly, B2 - 4AC =
4 - (-96) =
100
Applying the quadratic formula :
2 ± √ 100
n = —————
2
Can 100 be simplified ?
Yes! The prime factorization of 100 is
2•2•5•5
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
100 = √2•2•5•5 =2•5•√ 1 =
± 10 • √ 1 =
± 10

So now we are looking at:
n = ( 2 ± 10) / 2
Two real solutions:
n =(2+√100)/2=1+5= 6.000
or:
n =(2-√100)/2=1-5= -4.000

Two solutions were found :

Text 224444

  1. n = 6
  2. n = -4

Textnut 2.24 Meters


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