Question

solve x^4-34x^2=-225

Answer 1

let x^2 be t

then the eqn. becomes:
t^2-34t+225=0

now solve the quadratic and get values of t, and eliminate nay negative values of t, since t is x^2, which cannot be negative.

Answer 2

\[x ^{4}-34x ^{2}=-225\]First, let's set this equal to 0.

\[x ^{4}-34x ^{2}+225=0\]
From there, examine the terms and identify it as being quadratic-like because it has three terms that mimic a quadratic (as the power of x on the second term is half the power of x on the first term, and the third term has no x).

Then, as others have said, we want to make a substitution:
\[u=x ^{2}\]
In order to sub it in for the first term, we can do a little re-write first.
\[(x ^{2})^2-34x ^{2}+225=0\]
Now you can sub it in:
\[(u)^2-34(u)+225=0\]
Now, factor like usual:
\[(u-25)(u-9)=0\]
And then use the Zero-Product Property:
\[u-25=0\]
\[u-9=0\]Solve for u

u = 9 and u = 25

At this point, don't forget, you were solving for x to begin with, so now you have to substitute back.
\[u=x ^{2}\]Therefore:
\[x ^{2}=9\]and \[x ^{2}=25\]

To solve for x, take the square root of both sides for each equation.
\[\sqrt{x ^{2}}=\pm \sqrt{9}\]\[x=\pm3\]
and
\[\sqrt{x ^{2}}=\pm \sqrt{25}\]\[x=\pm5\]