# How to split large numbers easily for factoring the middle term?

I was doing a problem when I got suddenly stuck with a middle term factorisation.

The equation is as follows:

$$x^4-289x^2+14400=0.$$

I managed to somehow solve it, but it took a lot of time.

Can you please help me with a fast way if any, or can you at least tell me what is the fastest method you know?

Thank you.

• You could solve $y^2-289y+14400=0$ using the quadratic formula – J. W. Tanner Nov 6 '20 at 17:00
• Use the quadratic formula? It does the job for you with no hassle. Here $x^2 = (289 \pm \sqrt{289^2 - 4 \cdot 1 \cdot 14400}) / 2 \cdot 1$, $x^2 = 225$ or $x^2 = 64$. You do the rest. – vonbrand Nov 6 '20 at 17:02
• Interesting: poshenloh.com/quadraticdetail – Wolgwang Nov 6 '20 at 17:04
• How did you solve it? – J. W. Tanner Nov 6 '20 at 17:08

A direct method would be to let $$y=x^2$$,
and solve $$y^2-289y+14400=0$$ using the quadratic formula.
• Indeed, any quartic equation of the form $x^4+bx^2+c=0$ should be solved this way. It is often said that this is a 'quadratic in $x^2$', just as $y^2+by+c=0$ is a quadratic in $y$. You can consider $x^2$ term to be the unknown that needs to be solved for, and then you can compute $x$ by taking the positive and negative square root of $x^2$. – Joe Nov 6 '20 at 21:16