Question

Solve 9 (X-5)=8(x+5) X=?

Answer 1

X is going to be equal to 85

Answer 2

Answer:

=(31+√-479)/-16=31/-16-i/16√ 479 = -1.9375+1.3679i

Rearrange:

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

                    9*(x-5)-(8*(x+5)*x)=0  

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 (9 • (x - 5)) -  (8 • (x + 5) • x)  = 0  

Step  2  :

Equation at the end of step  2  :

 (9 • (x - 5)) -  8x • (x + 5)  = 0  

Step  3  :

Equation at the end of step  3  :

 9 • (x - 5) -  8x • (x + 5)  = 0  

Step  4  :

Step  5  :

Pulling out like terms :

5.1     Pull out like factors :

  -8x2 - 31x - 45  =   -1 • (8x2 + 31x + 45)  

Trying to factor by splitting the middle term

5.2     Factoring  8x2 + 31x + 45  

The first term is,  8x2  its coefficient is  8 .

The middle term is,  +31x  its coefficient is  31 .

The last term, "the constant", is  +45  

Step-1 : Multiply the coefficient of the first term by the constant   8 • 45 = 360  

Step-2 : Find two factors of  360  whose sum equals the coefficient of the middle term, which is   31 .

     -360    +    -1    =    -361  

     -180    +    -2    =    -182  

     -120    +    -3    =    -123  

     -90    +    -4    =    -94  

     -72    +    -5    =    -77  

     -60    +    -6    =    -66  

For tidiness, printing of 42 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Equation at the end of step  5  :

 -8x2 - 31x - 45  = 0  

Step  6  :

Parabola, Finding the Vertex :

6.1      Find the Vertex of   y = -8x2-31x-45

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens down and accordingly has a highest point (AKA absolute maximum) .    We know this even before plotting  "y"  because the coefficient of the first term, -8 , is negative (smaller 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  -1.9375  

Plugging into the parabola formula  -1.9375  for  x  we can calculate the  y -coordinate :  

 y = -8.0 * -1.94 * -1.94 - 31.0 * -1.94 - 45.0

or   y = -14.969

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -8x2-31x-45

Axis of Symmetry (dashed)  {x}={-1.94}  

Vertex at  {x,y} = {-1.94,-14.97}  

Function has no real roots

Solve Quadratic Equation by Completing The Square

6.2     Solving   -8x2-31x-45 = 0 by Completing The Square .

Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:

8x2+31x+45 = 0  Divide both sides of the equation by  8  to have 1 as the coefficient of the first term :

  x2+(31/8)x+(45/8) = 0

Subtract  45/8  from both side of the equation :

  x2+(31/8)x = -45/8

Now the clever bit: Take the coefficient of  x , which is  31/8 , divide by two, giving  31/16 , and finally square it giving  961/256  

Add  961/256  to both sides of the equation :

 On the right hand side we have :

  -45/8  +  961/256   The common denominator of the two fractions is  256   Adding  (-1440/256)+(961/256)  gives  -479/256  

 So adding to both sides we finally get :

  x2+(31/8)x+(961/256) = -479/256

Adding  961/256  has completed the left hand side into a perfect square :

  x2+(31/8)x+(961/256)  =

  (x+(31/16)) • (x+(31/16))  =

 (x+(31/16))2

Things which are equal to the same thing are also equal to one another. Since

  x2+(31/8)x+(961/256) = -479/256 and

  x2+(31/8)x+(961/256) = (x+(31/16))2

then, according to the law of transitivity,

  (x+(31/16))2 = -479/256

We'll refer to this Equation as  Eq. #6.2.1  

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

Note that the square root of

  (x+(31/16))2   is

  (x+(31/16))2/2 =

 (x+(31/16))1 =

  x+(31/16)

Now, applying the Square Root Principle to  Eq. #6.2.1  we get:

  x+(31/16) = √ -479/256

Subtract  31/16  from both sides to obtain:

  x = -31/16 + √ -479/256

In Math,  i  is called the imaginary unit. It satisfies   i2  =-1. Both   i   and   -i   are the square roots of   -1  

Since a square root has two values, one positive and the other negative

  x2 + (31/8)x + (45/8) = 0

  has two solutions:

 x = -31/16 + √ 479/256 •  i  

  or

 x = -31/16 - √ 479/256 •  i  

Note that  √ 479/256 can be written as

 √ 479  / √ 256   which is √ 479  / 16

Solve Quadra

Step-by-step explanation: