X^2 15x

X^2 15x DEFAULT

Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2". 

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     x^2-15*x-(2)=0 

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x2-15x-2 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -15x  its coefficient is  -15 .
The last term, "the constant", is  -2 

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   -15 .

     -2   +   1   =   -1
     -1   +   2   =   1


Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  1  :

x2 - 15x - 2 = 0

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = x2-15x-2

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   7.5000  

 Plugging into the parabola formula   7.5000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 7.50 * 7.50 - 15.0 * 7.50 - 2.0
or   y = -58.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x2-15x-2
Axis of Symmetry (dashed)  {x}={ 7.50} 
Vertex at  {x,y} = { 7.50,-58.25} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.13, 0.00} 
Root 2 at  {x,y} = {15.13, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   x2-15x-2 = 0 by Completing The Square .

 Add  2  to both side of the equation :
   x2-15x = 2

Now the clever bit: Take the coefficient of  x , which is  15 , divide by two, giving  15/2 , and finally square it giving  225/4 

Add  225/4  to both sides of the equation :
  On the right hand side we have :
   2  +  225/4    or,  (2/1)+(225/4) 
  The common denominator of the two fractions is  4   Adding  (8/4)+(225/4)  gives  233/4 
  So adding to both sides we finally get :
   x2-15x+(225/4) = 233/4

Adding  225/4  has completed the left hand side into a perfect square :
   x2-15x+(225/4)  =
   (x-(15/2)) • (x-(15/2))  =
  (x-(15/2))2
Things which are equal to the same thing are also equal to one another. Since
   x2-15x+(225/4) = 233/4 and
   x2-15x+(225/4) = (x-(15/2))2
then, according to the law of transitivity,
   (x-(15/2))2 = 233/4

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.

Note that the square root of
   (x-(15/2))2  is
   (x-(15/2))2/2 =
  (x-(15/2))1 =
   x-(15/2)

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   x-(15/2) = √ 233/4

Add  15/2  to both sides to obtain:
   x = 15/2 + √ 233/4

Since a square root has two values, one positive and the other negative
   x2 - 15x - 2 = 0
   has two solutions:
  x = 15/2 + √ 233/4
   or
  x = 15/2 - √ 233/4

Note that  √ 233/4 can be written as
  √ 233  / √ 4   which is √ 233  / 2

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    x2-15x-2 = 0 by the Quadratic Formula .

 According 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

  In our case,  A   =     1
                      B   =   -15
                      C   =   -2

Accordingly,  B2  -  4AC   =
                     225 - (-8) =
                     233

Applying the quadratic formula :

               15 ± √ 233
   x  =    ——————
                      2

  √ 233   , rounded to 4 decimal digits, is  15.2643
 So now we are looking at:
           x  =  ( 15 ±  15.264 ) / 2

Two real solutions:

 x =(15+√233)/2=15.132

or:

 x =(15-√233)/2=-0.132

Two solutions were found :

  1.  x =(15-√233)/2=-0.132
  2.  x =(15+√233)/2=15.132
Sours: https://www.tiger-algebra.com/drill/x2-15x=2/
\left(x-9\right)\left(x-6\right)
Tick mark Image
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+54. To find a and b, set up a system to be solved.
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 54.
-1,-54 -2,-27 -3,-18 -6,-9
Calculate the sum for each pair.
-1-54=-55 -2-27=-29 -3-18=-21 -6-9=-15
The solution is the pair that gives sum -15.
Rewrite x^{2}-15x+54 as \left(x^{2}-9x\right)+\left(-6x+54\right).
\left(x^{2}-9x\right)+\left(-6x+54\right)
Factor out x in the first and -6 in the second group.
x\left(x-9\right)-6\left(x-9\right)
Factor out common term x-9 by using distributive property.
\left(x-9\right)\left(x-6\right)
\left(x-9\right)\left(x-6\right)
Tick mark Image

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Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2". 

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x2-15x-500 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -15x  its coefficient is  -15 .
The last term, "the constant", is  -500 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -500 = -500 

Step-2 : Find two factors of  -500  whose sum equals the coefficient of the middle term, which is   -15 .

     -500   +   1   =   -499
     -250   +   2   =   -248
     -125   +   4   =   -121
     -100   +   5   =   -95
     -50   +   10   =   -40
     -25   +   20   =   -5
     -20   +   25   =   5
     -10   +   50   =   40
     -5   +   100   =   95
     -4   +   125   =   121
     -2   +   250   =   248
     -1   +   500   =   499


Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  1  :

x2 - 15x - 500 = 0

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = x2-15x-500

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   7.5000  

 Plugging into the parabola formula   7.5000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 7.50 * 7.50 - 15.0 * 7.50 - 500.0
or   y = -556.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x2-15x-500
Axis of Symmetry (dashed)  {x}={ 7.50} 
Vertex at  {x,y} = { 7.50,-556.25} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-16.08, 0.00} 
Root 2 at  {x,y} = {31.08, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   x2-15x-500 = 0 by Completing The Square .

 Add  500  to both side of the equation :
   x2-15x = 500

Now the clever bit: Take the coefficient of  x , which is  15 , divide by two, giving  15/2 , and finally square it giving  225/4 

Add  225/4  to both sides of the equation :
  On the right hand side we have :
   500  +  225/4    or,  (500/1)+(225/4) 
  The common denominator of the two fractions is  4   Adding  (2000/4)+(225/4)  gives  2225/4 
  So adding to both sides we finally get :
   x2-15x+(225/4) = 2225/4

Adding  225/4  has completed the left hand side into a perfect square :
   x2-15x+(225/4)  =
   (x-(15/2)) • (x-(15/2))  =
  (x-(15/2))2
Things which are equal to the same thing are also equal to one another. Since
   x2-15x+(225/4) = 2225/4 and
   x2-15x+(225/4) = (x-(15/2))2
then, according to the law of transitivity,
   (x-(15/2))2 = 2225/4

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.

Note that the square root of
   (x-(15/2))2  is
   (x-(15/2))2/2 =
  (x-(15/2))1 =
   x-(15/2)

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   x-(15/2) = √ 2225/4

Add  15/2  to both sides to obtain:
   x = 15/2 + √ 2225/4

Since a square root has two values, one positive and the other negative
   x2 - 15x - 500 = 0
   has two solutions:
  x = 15/2 + √ 2225/4
   or
  x = 15/2 - √ 2225/4

Note that  √ 2225/4 can be written as
  √ 2225  / √ 4   which is √ 2225  / 2

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    x2-15x-500 = 0 by the Quadratic Formula .

 According 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

  In our case,  A   =     1
                      B   =   -15
                      C   =  -500

Accordingly,  B2  -  4AC   =
                     225 - (-2000) =
                     2225

Applying the quadratic formula :

               15 ± √ 2225
   x  =    ——————
                      2

Can  √ 2225 be simplified ?

Yes!   The prime factorization of  2225   is
   5•5•89 
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).

√ 2225   =  √ 5•5•89   =
                ±  5 • √ 89

  √ 89   , rounded to 4 decimal digits, is   9.4340
 So now we are looking at:
           x  =  ( 15 ± 5 •  9.434 ) / 2

Two real solutions:

 x =(15+√2225)/2=(15+5√ 89 )/2= 31.085

or:

 x =(15-√2225)/2=(15-5√ 89 )/2= -16.085

Two solutions were found :

  1.  x =(15-√2225)/2=(15-5√ 89 )/2= -16.085
  2.  x =(15+√2225)/2=(15+5√ 89 )/2= 31.085
Sours: https://www.tiger-algebra.com/drill/x2-15x-500=0/

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Solution for x^2-15x-2=0 equation:


Simplifying x2 + -15x + -2 = 0 Reorder the terms: -2 + -15x + x2 = 0 Solving -2 + -15x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + -15x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + -15x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -15x + x2 = 0 + 2 -15x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 -15x + x2 = 2 The x term is -15x. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15x + 56.25 + x2 = 2 + 56.25 Reorder the terms: 56.25 + -15x + x2 = 2 + 56.25 Combine like terms: 2 + 56.25 = 58.25 56.25 + -15x + x2 = 58.25 Factor a perfect square on the left side: (x + -7.5)(x + -7.5) = 58.25 Calculate the square root of the right side: 7.632168761 Break this problem into two subproblems by setting (x + -7.5) equal to 7.632168761 and -7.632168761.

Subproblem 1

x + -7.5 = 7.632168761 Simplifying x + -7.5 = 7.632168761 Reorder the terms: -7.5 + x = 7.632168761 Solving -7.5 + x = 7.632168761 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = 7.632168761 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 7.632168761 + 7.5 x = 7.632168761 + 7.5 Combine like terms: 7.632168761 + 7.5 = 15.132168761 x = 15.132168761 Simplifying x = 15.132168761

Subproblem 2

x + -7.5 = -7.632168761 Simplifying x + -7.5 = -7.632168761 Reorder the terms: -7.5 + x = -7.632168761 Solving -7.5 + x = -7.632168761 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = -7.632168761 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -7.632168761 + 7.5 x = -7.632168761 + 7.5 Combine like terms: -7.632168761 + 7.5 = -0.132168761 x = -0.132168761 Simplifying x = -0.132168761

Solution

The solution to the problem is based on the solutions from the subproblems. x = {15.132168761, -0.132168761}

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x²-15x+54=0 - Practice set 2.2 - 10th Maths - online education - class 10 educational video

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