Solve equation after making an appropriate substitution:

Question

X⁴ - 19x²+48=0 

Answer 1

x⁴ - 19x² + 48 = 0

 

(x²)² - 19(x²) + 48 = 0

 

Let u = x²

 

Then, u² - 19u + 48 = 0

 

        (u - 16)(u - 3) = 0

 

u = 16 or u = 3

 

So, x² = 16 or x² = 3

 

x = ±√16 or ±√3

 

x = 4, -4, √3, -√3

Answer 2

This is really similar to your standard binomial equation: ax²+bx+c.  We treat it similarly. and start by factoring 

 

Factor:

(x²-16)(x²-3)

 

We ask ourselves: can I factor THESE any further.  Yes I can, so set each equation to 0 and solve again

x2-16 = (x+4)(x-4); x = ±4

x2-3 = (x+√3)(x-√3); x = ±√3

 

He shoots, he scores, he gets the answer!  The crowd goes wild....