What are the roots of the equation x² 10x 2 0?

Our goal is to solving

First, move the

We need a constant term on the left side that makes the whole expression a perfect square. Take a half of

Add

Factoring

Solving for

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  2x²+12=16(Completing the Square method)

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  What do you call the value of x that is true to the equation but is not valid as a

  solution to the given problem​

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  4x²-28=0(Extracting Square Root)HELP PO

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 1.1     Factoring  x2-10x+2 

The first term is,  x2  its coefficient is  1 .

Step-1 : Multiply the coefficient of the first term by the constant   1 • 2 = 2 

Step-2 : Find two factors of  2  whose sum equals the coefficient of the middle term, which is   -10 .

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Equation at the end of step  1  :

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   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,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   5.0000 Plugging into the parabola formula   5.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 5.00 * 5.00 - 10.0 * 5.00 + 2.0
or   y = -23.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x2-10x+2
Axis of Symmetry (dashed)  {x}={ 5.00} 
Vertex at  {x,y} = { 5.00,-23.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.20, 0.00} 
Root 2 at  {x,y} = { 9.80, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   x2-10x+2 = 0 by Completing The SquareSubtract  2  from both side of the equation :
   x2-10x = -2

Now the clever bit: Take the coefficient of  x , which is  10 , divide by two, giving  5 , and finally square it giving  25 

Add  25  to both sides of the equation :

   -2  +  25    or,  (-2/1)+(25/1) 

   x2-10x+25 = 23

Adding  25  has completed the left hand side into a perfect square :

   x2-10x+25 = 23 and

   (x-5)2 = 23

We'll refer to this Equation as  Eq. #2.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

   (x-5)2   is

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:

Add  5  to both sides to obtain:

   x2 - 10x + 2 = 0

  x = 5 + √ 23

  x = 5 - √ 23

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    x2-10x+2 = 0 by the Quadratic FormulaAccording to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :

                                     
            - B  ±  √ B2-4AC  x =   ————————                      2A

               Can  √ 92 be simplified ?

Yes!   The prime factorization of  92   is

√ 92   =  √ 2•2•23   =
                ±  2 • √ 23

  √ 23   , rounded to 4 decimal digits, is   4.7958
 So now we are looking at:

 x =(10+√92)/2=5+√ 23 = 9.796

 x =(10-√92)/2=5-√ 23 = 0.204

Two solutions were found :

1.  x =(10-√92)/2=5-√ 23 = 0.204
2.  x =(10+√92)/2=5+√ 23 = 9.796