# Same equation, different results (Quartic function)

I've encountered this equation:

$-5x(1+4x^2)^{-3/2}+{1\over\sqrt{1+4x^2}}=0$

I always use Cymath and Symbolab calculators, but this time they give different answers to the same equation.

Symbolab result

Cymath result

I got the same result as Symbolab but by a different approach. I solve it by replacing the $x$ with $x^2$ and solving it as a quadratic function. (I hope you understand what I mean, not good at explaining things in english)

Questions:

• Which calculator is the correct?
• Is there any better calculator?

Bonus question: What is the point 9 doing in Cymath result? Any links that explains this property? I've been trying for hours to find out what it does...

I reach a point where I'm left with: $16x^4-17x^2+1$ Then I do $u = x^2$

So: $16u^2-17u+1$

Then I do this:

$-b\pm\sqrt{b^2-4ac}\over 2a$

And I get the results 1 and $1\over16$

So:

$x^2 = 1$ -> $x=\sqrt{1}$ -> $x=1$

$x^2 = {1\over16}$ -> $x=\sqrt{{1\over16}}$ -> $x={\sqrt{1}\over\sqrt{16}}$ -> $x={1\over4}$

• In Cymath solution, look for the result after step 8, they suddenly reduced the order of the polynomial by 2. So they solved for $x^2$ instead of $x$. Just like you did. But afterwards when checking answers they didn't use that fact. – Kaster Nov 6 '16 at 2:20
• As for your second question about step 9, well they used some intuition on how to simplify a polynomial with integer coefficients. My advice, unless you really see what's going on in there in terms of roots and such (see Vieta's forumlas), don't do that. You'll spend more time doing it. Instead use some more practical methods, like polynomial division or exact root formulas and etc. – Kaster Nov 6 '16 at 2:24
• @Kaster thank you very much! I was trying to find out what the ... the computer was doing in that step! – facumedica Nov 6 '16 at 2:29
• It is trying to factor the quadratic polynomial empirically, as I explained in my answer below. – Momo Nov 6 '16 at 2:30

Symbolab does it OK. Cymath changes the equation $16x^4-17x^2+1$ into $16x^2-17x+1$ between the steps 8 and 9 (software bug), so the answer is wrong.
At step 9, I guess Cymath is trying to write $16x^2-17x+1=(ax+1)(bx+1)=abx^2+(a+b)x+1$ and work out $a$ and $b$