Question

solve for q in 4000 + 10q^2-q^3
show calculation.

Answer 1

Arrange the equation,

q³-10q²=4000

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

q^3-10*q^2-(4000)=0

Now,

((q3) -  (2•5q2)) -  4000  = 0 

(q2 + 10q + 200) • (q - 20)  = 0 

Solving    q²+10q+200 = 0 by the Quadratic Formula .

According to the Quadratic Formula,  q  , the solution for   Aq²+Bq+C  = 0  , where  A, B and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ± ? B²-4AC
  q =   ————————
                      2A

  In our case,  A   =     1
                      B   =    10
                      C   =  200

Accordingly,  B²  -  4AC   =
                     100 - 800 =
                     -700

Applying the quadratic formula :

               -10 ± ? -700
   q  =    ——————
                      2
Both   i   and   -i   are the square roots of minus 1

Accordingly,? -700  =
                    ? 700 • (-1)  =
                    ? 700  • ? -1   =
                    ±  ? 700  • i
? 700   =  ? 2•2•5•5•7   =2•5•? 7   =
                ±  10 • ? 7
  

? 7   , rounded to 4 decimal digits, is   2.6458
So now we are looking at:
           q  =  ( -10 ± 10 • 2.646 i ) / 2

Two imaginary solutions :

 q =(-10+?-700)/2=-5+5i? 7 = -5.0000+13.2288i
  or: 
 q =(-10-?-700)/2=-5-5i? 7 = -5.0000-13.2288i

Now, solving single variable equation-

q-20 = 0

Add  20 to both sides of the equation :
                      q = 20

Hence, Three solutions were found :

1. q = 20
2. q =(-10-?-700)/2=-5-5i? 7 = -5.0000-13.2288i
3. q =(-10+?-700)/2=-5+5i? 7 = -5.0000+13.2288i