Solve x^4 + x^2 – 2 = 0. (MULTIPLE CHOICE)

Question

Solve x⁴ + x² – 2 = 0.

 

a. x = ±1 or x = ±i
b. x = ±1 or x = i√2
c. x = ±1 or x = ±2
d. x = ±1 or x = √2

 

Choose one thanks

Answer 1

Factor

 

( x² +2) (x² - 1) = 0

 

x² + 2 = 0       or     x² - 1 ⁼ 0

 

(x + i√2) ( x - i√2) =0      or    (x + 1) (x-1) = 0

 

x + i√2 = 0   x - i√2 = 0    or   x+1 = 0   x-1 = 0

 

x= -i√2           x = i√2               x = -1      x = 1

Answer 2

Let u = x².  Then the equation becomes u² + u - 2 = 0.

 

Factoring, we get (u + 2)(u - 1) = 0

 

So, u = -2 or u = 1

 

Since u = x², x²= -2 or x² = 1.

 

If x² = 1, then x = ±1.

 

If x² = -2, then x = ±√(-2) = ±i√2

 

Solutions:  1, -1, i√2, -i√2

Answer 3

Let y = x².  Then, y² = (x²)² = x⁴.

Thus, the equation

x⁴ + x² - 2 = 0

becomes

y² + y - 2 = 0.

We can factor this as (y+2)(y-1) = 0, which has solutions y = -2 and y = 1.

Now, recall that we made the substitution y = x².

Thus, we have x² = -2 or x² = 1.

Thus, from x² = -2, we get x = ±i√2, and from x² = 1, we get x = ±1

Then, the solution is b.

Answer 4

b

jim