Find all (real and imaginary) solutions to the polynomial equations by factoring and or using the quadratic formula.

Question

X^4+9=-6x^2

Answer 1

To "solve" an equation means to find all the values for which it is true.  Let's find all the solutions to:

      x⁴ + 9 = -6x²

 

Typically, polynomials are written "=0" when we solve them.  So:

      x⁴ + 6x² +9 = 0

 

Now, advanced math builds on easy math (that is, it has prerequisites), so you will see many patterns over and over with more sophistication.  Remember F-O-I-L (First-Inside-Outside-Last)?  One special case of F-O-I-L uses the square of values:

       (y+a)(y+a)  =  y² + 2ay + a²

 

Now, we learn about "substitution."  It is a cool way to change how an expression looks so we will recognize how to solve it.  Let y=x² and a=3:

            (y+3)(y+3)  =  y² + 6y + 9  =  0

      but y=x², so

            (x²+3)(x²+3) =  x⁴ + 6x² + 9 = 0

 

Now, we apply the Multiplicative Property of Zero:    if a*b=0, then either a=0 or b=0 or both.

 

That means:

         x²+3 = 0             [note: you could write twice]

          x²=-3

          x= ±(√3)i

 

Check:

   Is   (+(√3)i)⁴ + 9 = -6(+(√3)i)²   ?

          9         + 9 =  -6(-3)   ?

                  18  =  18   ?yes

   Is   (-(√3)i)⁴ + 9 = -6(-(√3)i)²   ?

             9       + 9  =  -6(-3)   ?

                  18  =  18   ?yes