SOLUTION: 2. The following equation has degree 4 because the highest power of the variable x is 4: x^4 – 3 x^2 + 2 = 0. Explain how one can use the quadratic formula to solve this equation

Algebra ->  Equations -> SOLUTION: 2. The following equation has degree 4 because the highest power of the variable x is 4: x^4 – 3 x^2 + 2 = 0. Explain how one can use the quadratic formula to solve this equation      Log On


   

Question 122287: 2. The following equation has degree 4 because the highest power of the variable x is 4:
x^4 – 3 x^2 + 2 = 0.
Explain how one can use the quadratic formula to solve this equation by using a change of variable. Solve this equation completely.
Found 2 solutions by josmiceli, solver91311:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E4+%96+3x%5E2+%2B+2+=+0
z+=+x%5E2
z%5E2+-+3z+%2B+2+=+0
%28z+-+2%29%28z+-+1%29+=+0
z+=+2
z+=+1
x%5E2+=+2
x%5E2+=+1
x+=+0+%2B-sqrt%282%29
x+=+0+%2B-sqrt%281%29
The solutions are:
x+=+sqrt%282%29
x+=+-sqrt%282%29
x+=+1
x+=+-1

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!






Let t=x%5E2, which means that t%5E2=x%5E4





Now substitute:











t%5E2-3t%2B2=0 is factorable





%28t-1%29%28t-2%29=0 => t=1 or t=2, so you really don't need the

quadratic formula to solve it.  But since the question asked for it, here

it is



a = 1

b = -3

c = 2



t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ 

t+=+%28-%28-3%29+%2B-+sqrt%28+%28-3%29%5E2-4%2A%281%29%2A%282%29+%29%29%2F%282%2A%281%29%29+ 

t+=+%283+%2B-+sqrt%28+9-8+%29%29%2F2 => t=%283%2B1%29%2F2=4%2F2=2 or t=%283-1%29%2F2=2%2F2=1



And no surprise, the roots are still 1 and 2.



 

But remember t=x%5E2





If t=1 then x=%2Bsqrt%281%29 or x=-sqrt%281%29 => x=1 or x=-1



If t=2 then x=%2Bsqrt%282%29 or x=-sqrt%282%29





This gives us a total of four roots for the original quartic (degree 4)

equation as the Fundamental Theorem of Algebra would lead us to suspect.





Just for fun, let's check the notion graphically:





graph%28600%2C600%2C-5%2C5%2C-5%2C5%2Cx%5E4-3x%5E2%2B2%29





Note that the graph intersects the x-axis in 4 places and the x-coordinates of

these 4 places are equal to the roots or zeros of the given equation.



How very tidy.