Question

What are the real or imaginary solutions of the polynomial equation? x^4 -52x^2 + 576= 0
A. 4, -4
B. 4, -6
c. 4, -4, 6, -6
d. no solution

Answer 1

u could factor the equation like this
\[\large x^4-52x^2+576=(x^2-a)(x^2-b) \]
such that \(a\cdot b=576\) and \(a+b=52\)

Answer 2

Another way (maybe easier), would be to solve by substitution:
Let \[u=x^{2}\]
then
\[x ^{4}-52x^2+576 = u^2-52u+576\]
Then use the quadratic formula:
\[u= \frac{ 52 \pm \sqrt{(-52)^2-4*1*576}}{ 2 }\]
\[u=36, 16\]
Substitute back in for u:
\[u = x^2 = 36, 16\]
\[x = \pm6, \pm4\]