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Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "q3"   was replaced by   "q^3".  1 more similar replacement(s).


(2): "1.5" was replaced by "(15/10)".

Rearrange:

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

           t*c-(1500+150*q-15*q^2+(15/10)*q^3)=0 

Step  1  :

            3
 Simplify   —
            2

Equation at the end of step  1  :

                               3
  tc-(((150q+1500)-(15•(q2)))+(—•q3))  = 0 
                               2

Step  2  :

Equation at the end of step  2  :

                                           3q3
  tc -  (((150q + 1500) -  (15 • (q2))) +  ———)  = 0 
                                            2 
 Step  3  :

Equation at the end of step  3  :

                                       3q3
  tc -  (((150q + 1500) -  (3•5q2)) +  ———)  = 0 
                                        2 

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a fraction to a whole

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

                            -15q2 + 150q + 1500     (-15q2 + 150q + 1500) • 2
     -15q2 + 150q + 1500 =  ———————————————————  =  —————————————————————————
                                     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

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   -15q² + 150q + 1500  =   -15 • (q² - 10q - 100) 

Trying to factor by splitting the middle term

 5.2     Factoring  q² - 10q - 100 

The first term is,  q²  its coefficient is  1 .
The middle term is,  -10q  its coefficient is  -10 .
The last term, "the constant", is  -100 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -100 = -100 

Step-2 : Find two factors of  -100  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

Adding fractions that have a common denominator :

 5.3       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:

 -15 • (q2-10q-100) • 2 + 3q3     3q3 - 30q2 + 300q + 3000
 ————————————————————————————  =  ————————————————————————
              2                              2            

Equation at the end of step  5  :

        (3q3 - 30q2 + 300q + 3000)
  tc -  ——————————————————————————  = 0 
                    2             

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a fraction from a whole

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

           tc     tc • 2
     tc =  ——  =  ——————
           1        2   

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   3q³ - 30q² + 300q + 3000  = 

  3 • (q³ - 10q² + 100q + 1000) 

Checking for a perfect cube :

 7.2    q³ - 10q² + 100q + 1000  is not a perfect cube

Trying to factor by pulling out :

 7.3      Factoring:  q³ - 10q² + 100q + 1000 

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

Group 1:  q³ - 10q² 
Group 2:  100q + 1000 

Pull out from each group separately :

Group 1:   (q - 10) • (q²)
Group 2:   (q + 10) • (100)

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 :

 7.4    Find roots (zeroes) of :       F(q) = q³ - 10q² + 100q + 1000
Polynomial Roots Calculator is a set of methods aimed at finding values of  q  for which   F(q)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  q  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  1  and the Trailing Constant is  1000.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,5 ,8 ,10 ,20 ,25 ,40 ,50 , etc

 Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 tc • 2 - (3 • (q3-10q2+100q+1000))     2tc - 3q3 + 30q2 - 300q - 3000
 ——————————————————————————————————  =  ——————————————————————————————
                 2                                    2               

Equation at the end of step  7  :

  2tc - 3q3 + 30q2 - 300q - 3000
  ——————————————————————————————  = 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:

  2tc-3q3+30q2-300q-3000
  —————————————————————— • 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 :
   2tc-3q³+30q²-300q-3000  = 0

Solving a Single Variable Equation :

 8.2     Solve   2tc-3q³+30q²-300q-3000  = 0

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.

We shall not handle this type of equations at this time.

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