SOLUTION: Solve x^4 +4x^2 = 32. Find 4 roots. Thanks.

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Question 1013108: Solve x^4 +4x^2 = 32. Find 4 roots.
Thanks.
Found 4 solutions by fractalier, ikleyn, macston, josmiceli:
Answer by fractalier(6550) About Me  (Show Source):
You can put this solution on YOUR website!
Just did this for you...
Okay, from
x^4 + 4x^2 = 32
x^4 + 4x^2 - 32 = 0
we can factor and get
(x^2 + 8)(x^2 - 4) = 0
(x + 2isqrt(2))(x - 2isqrt(2))(x + 2)(x - 2) = 0
giving us four roots
x = +/- 2isqrt(2) and x = +/- 2

Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve x^4 +4x^2 = 32. Find 4 roots.
----------------------------------------- 

x%5E4+%2B4x%5E2 = 32,  

%28x%5E2%2B2%29%5E2 - 4 = 32,

%28x%5E2%2B2%29%5E2 = 36,

%28x%5E2%2B2%29 = +/- 6.

Now solve two quadratic equations 

1. x%5E2%2B2 = 6  ---->  x%5E2 = 4  ---->  Two solutions: x = 2 and x = -2.

2. x%5E2%2B2 = -6  ---->  x%5E2 = -8  ---->  Two solutions: x = 2i%2Asqrt%282%29 and x = -2i%2Asqrt%282%29.

You obtain four solutions, in total.


Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let y=x^2
.
x^4+4x^2=32
x^4+4x^2-32=0 . Substitute y for x^2
y^2+4y-32=0
(y+8)(y-4)=0
y+8=0 --OR-- y-4=0
y=-8 --OR-- y=4 . Replace y with x^2
x^2=-8 --or-- x^2=4
x=+/-sqrt%28-8%29 --OR-- x=+/-2
x=+/-2sqrt%282%29i --OR-- x=+/-2



Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +z+=+x%5E2+
Substitute +z+ in the equation
+z%5E2+%2B+4z+=+32+
Complete the square:
+z%5E2+%2B+4z+%2B+%28+4%2F2+%29%5E2+=+32+%2B+%28+4%2F2+%29%5E2+
+z%5E2+%2B+4z+%2B+4+=+32+%2B+4+
+%28+z%2B+2+%29%5E2+=+6%5E2+
Take the square root of both sides
+z+%2B+2+=+6+
+z+=+4%0D%0Aand+also%0D%0A%7B%7B%7Bz+%2B+2+=+-6+
+z+=+-8+
-----------------
+x%5E2+=+z+
+x%5E2+=+4+
+x+=+2+
+x+=+-2+
-------------
+x%5E2+=+-8+
+x+=+sqrt%28-8%29+
+x+=+2%2Asqrt%282%29%2Ai+
and
+x+=+-sqrt%28-8%29+
+x+=+-2%2Asqrt%282%29%2Ai+
-------------------
The 4 roots are:
+x+=+2+
+x+=+-2+
+x+=+2%2Asqrt%282%29%2Ai+
+x+=+-2%2Asqrt%282%29%2Ai+
---------------------
You can plug these back into the
equation to check