How do you solve 4x^4 - 65x^2 + 16 with all the possible solutions. I tried graphing it, but don't know how

Question

to get the solution from the graph

Answer 1

The thing to notice is that you only have even powers, so what I would recommend is replacing x² with k.

Now your equation is:
4k² - 65k + 16 = 0

You can solve this easily with factoring, or the quadratic equation.

k² - (65/4)k + 4 = 0

(k - 16)(k - 1/4) = 0

k = 16 or k = 1/4

At this point, plug x² back into the solutions:
x² = 16 or x² = 1/4

Don't forget the positive and negative square roots, so the answers are:
x = -4, +4, -1/2, 1/2

Answer 2

Ok Let's see if we can help.

4x^4 -65x^2 +16=0 Factorize
(4x^2 -1)(x^2-16)=0

So we solve and see that:
4x^2=1 or x^2=1/4 and x=+/- 1/2 is one set of answers.

Also x^2=16 so x= +/- 4 is another set of answers.

Let's do the check for one of the four

4(1/2)^4 - 65(1/2)^2 +16 =0 ??
4/16 - 65/4 +16 =0 ??
1/4 - 16 1/4 +16 =0??
-16+16=0 YES!!

Hope this helps!

Answer 3

that =0 write it out