Solution - Approximation

Step by step solution :

Step  1  :

            9
 Simplify   —
            2

Equation at the end of step  1  :

             9                    
  (((x3) +  (— • x2)) -  12x) +  5  = 0 
             2                    

Step  2  :

Equation at the end of step  2  :

            9x2             
  (((x3) +  ———) -  12x) +  5  = 0 
             2              
 Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  2  as the denominator :

           x3     x3 • 2
     x3 =  ——  =  ——————
           1        2   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 x3 • 2 + 9x2     2x3 + 9x2
 ————————————  =  —————————
      2               2    

Equation at the end of step  3  :

   (2x3 + 9x2)            
  (——————————— -  12x) +  5  = 0 
        2                 

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  2  as the denominator :

           12x     12x • 2
    12x =  ———  =  ———————
            1         2   

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   2x³ + 9x²  =   x² • (2x + 9) 

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 x2 • (2x+9) - (12x • 2)     2x3 + 9x2 - 24x
 ———————————————————————  =  ———————————————
            2                       2       

Equation at the end of step  5  :

  (2x3 + 9x2 - 24x)    
  ————————————————— +  5  = 0 
          2            

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  2  as the denominator :

         5     5 • 2
    5 =  —  =  —————
         1       2  

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   2x³ + 9x² - 24x  =   x • (2x² + 9x - 24) 

Trying to factor by splitting the middle term

 7.2     Factoring  2x² + 9x - 24 

The first term is,  2x²  its coefficient is  2 .
The middle term is,  +9x  its coefficient is  9 .
The last term, "the constant", is  -24 

Step-1 : Multiply the coefficient of the first term by the constant   2 • -24 = -48 

Step-2 : Find two factors of  -48  whose sum equals the coefficient of the middle term, which is   9 .

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

Adding fractions that have a common denominator :

 7.3       Adding up the two equivalent fractions

 x • (2x2+9x-24) + 5 • 2     2x3 + 9x2 - 24x + 10
 ———————————————————————  =  ————————————————————
            2                         2          

Checking for a perfect cube :

 7.4    2x³ + 9x² - 24x + 10  is not a perfect cube

Trying to factor by pulling out :

 7.5      Factoring:  2x³ + 9x² - 24x + 10 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -24x + 10 
Group 2:  2x³ + 9x² 

Pull out from each group separately :

Group 1:   (12x - 5) • (-2)
Group 2:   (2x + 9) • (x²)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 7.6    Find roots (zeroes) of :       F(x) = 2x³ + 9x² - 24x + 10
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  2  and the Trailing Constant is  10.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,2 ,5 ,10

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  7  :

  2x3 + 9x2 - 24x + 10
  ————————————————————  = 0 
           2          

Step  8  :

When a fraction equals zero :

 8.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  2x3+9x2-24x+10
  —————————————— • 2 = 0 • 2
        2       

Now, on the left hand side, the  2  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   2x³+9x²-24x+10  = 0

Cubic Equations :

 8.2     Solve   2x³+9x²-24x+10 = 0

Future releases of Tiger-Algebra will solve equations of the third degree directly.

Meanwhile we will use the Bisection method to approximate one real solution.

Approximating a root using the Bisection Method :

We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).

The function is   F(x) = 2x³ + 9x² - 24x + 10

At   x=   -7.00   F(x)  is equal to  -67.00 
At   x=   -6.00   F(x)  is equal to  46.00 

Intuitively we feel, and justly so, that since  F(x)  is negative on one side of the interval, and positive on the other side then, somewhere inside this interval,  F(x)  is zero

Procedure :
(1) Find a point "Left" where F(Left) < 0

(2) Find a point 'Right' where F(Right) > 0

(3) Compute 'Middle' the middle point of the interval [Left,Right]

(4) Calculate Value = F(Middle)

(5) If Value is close enough to zero goto Step (7)

Else :
If Value < 0 then : Left <- Middle
If Value > 0 then : Right <- Middle

(6) Loop back to Step (3)

(7) Done!! The approximation found is Middle

Follow Middle movements to understand how it works :

    Left       Value(Left)     Right       Value(Right)

-7.000000000  -67.000000000 -6.000000000   46.000000000
-7.000000000  -67.000000000  0.000000000   10.000000000
-7.000000000  -67.000000000 -3.500000000  118.500000000
-7.000000000  -67.000000000 -5.250000000   94.656250000
-7.000000000  -67.000000000 -6.125000000   35.074218750
-6.562500000  -10.148925781 -6.125000000   35.074218750
-6.562500000  -10.148925781 -6.343750000   13.853332520
-6.562500000  -10.148925781 -6.453125000    2.207725525
-6.507812500   -3.880738258 -6.453125000    2.207725525
-6.480468750   -0.814163566 -6.453125000    2.207725525
-6.480468750   -0.814163566 -6.466796875    0.702351347
-6.473632812   -0.054511601 -6.466796875    0.702351347
-6.473632812   -0.054511601 -6.470214844    0.324268260
-6.473632812   -0.054511601 -6.471923828    0.134965456
-6.473632812   -0.054511601 -6.472778320    0.040248713
-6.473205566   -0.007125997 -6.472778320    0.040248713
-6.473205566   -0.007125997 -6.472991943    0.016562720
-6.473205566   -0.007125997 -6.473098755    0.004718702
-6.473152161   -0.001203563 -6.473098755    0.004718702
-6.473152161   -0.001203563 -6.473125458    0.001757591
-6.473152161   -0.001203563 -6.473138809    0.000277019
-6.473145485   -0.000463270 -6.473138809    0.000277019
-6.473142147   -0.000093125 -6.473138809    0.000277019

     Next Middle will get us close enough to zero:

     F( -6.473141313 ) is  -0.000000589  

     The desired approximation of the solution is:

       x ≓ -6.473141313

     Note, ≓ is the approximation symbol

One solution was found :