# Thread: I need help finding the zeros of a function.

1. ## I need help finding the zeros of a function.

The function is:

$
4x^4+17x^2+4 = 0
$

I have tried to do the following:
$
x^2(4x^2+17) + 4 = 0
$

$
x^2(4x^2+17) = -4
$

$
x^2 = -4
$

$
x = 2i
$

$
4x^2+17 = -4
$

$
4x^2 = -21
$

$
x^2 = -21/4
$

$
x^2 = sqrt(-21)/2
$

But I know I do not have the right answer for there is not sqrt(-21)/2 as one of the answers in the back of the book. Any help would be appreciated.

2. let x^2 = w then x^4 = w^2

$4w^2+17w+4=0$

3. Originally Posted by thyrgle
The function is:

$
4x^4+17x^2+4 = 0
$

I have tried to do the following:
$
x^2(4x^2+17) + 4 = 0
$

$
x^2(4x^2+17) = -4
$

$
x^2 = -4
$

$
x = 2i
$

$
4x^2+17 = -4
$

$
4x^2 = -21
$

$
x^2 = -21/4
$

$
x^2 = sqrt(-21)/2
$

But I know I do not have the right answer for there is not sqrt(-21)/2 as one of the answers in the back of the book. Any help would be appreciated.
just after the line $x=2i$ how did you get $4x^2+17 = -4$??

4. $\displaystyle 4w^2+17w+4=0\Rightarrow w=-4, \ -\frac{1}{4}\Rightarrow x^2=-4, \ -\frac{1}{4}\Rightarrow x=\pm 2i, \ \pm\frac{i}{2}$

5. I get it know:

$
(w+4)(4w+1)
$

Then
$
(x^2+4) = 0 or (4x^2+1 = 0)
$

$
-2i, 2i, -1/2i, 1/2i
$

Thanks dwsmith!

6. It is true that if ab= 0, then either a= 0 or b= 0 but that is a property of 0 only!
Saying that $x^2(4x^2+17) = -4$ does NOT imply that " $x^2= -4$" or that " $4x^2+ 17= -4$".

As dwsmith said, you original equation is a quadratic in $w= x^2$. By the way, I would be inclined to write those last two roots as (1/2)i and (-1/2)i or even -i/2, i/2. -1/2i and 1/2i are too likely to be interpreted as -1/(2i) and 1/(2i) (which, it suddenly occurs to me, are exactly (1/2)i and (-1/2)i!)

7. Just following up on what HallsofIvy said, here is a specific counterexample:

$ab=-4$ has $a=2$, $b=-2$ as a solution. Notice that neither $a$ nor $b$ is $-4$.

Of course, there are infinitely many other solutions to this equation.