Is it possible to isolate the variable X in this equation?

Question

$P = \dfrac{1}{\sqrt{x^2+xac+a^2}} + \dfrac{1}{\sqrt{x^2+xbc+b^2}}$

Answer 1

Hint: Squaring your $P$ you will get $$P^2=\frac{1}{x^2+acx+a^2}+\frac{1}{x^2+bcx+b^2}+\frac{2}{\sqrt{x^2+acx+a^2}\sqrt{x^2+bcx+b^2}}$$. Now we can write $$P^2-\frac{1}{x^2+acx+a^2}-\frac{1}{x^2+bcx+b^2}=\frac{2}{\sqrt{x^2+acx+a^2}\sqrt{x^2+bcx+b^2}}$$ Now you must square again. Good luck! $${P}^{4}{a}^{2}{b}^{2}{c}^{4}{x}^{4}+2\,{P}^{4}{a}^{3}{b}^{2}{c}^{3}{x} ^{3}+2\,{P}^{4}{a}^{2}{b}^{3}{c}^{3}{x}^{3}+2\,{P}^{4}{a}^{2}b{c}^{3}{ x}^{5}+2\,{P}^{4}a{b}^{2}{c}^{3}{x}^{5}+{P}^{4}{a}^{4}{b}^{2}{c}^{2}{x }^{2}+4\,{P}^{4}{a}^{3}{b}^{3}{c}^{2}{x}^{2}+4\,{P}^{4}{a}^{3}b{c}^{2} {x}^{4}+{P}^{4}{a}^{2}{b}^{4}{c}^{2}{x}^{2}+4\,{P}^{4}{a}^{2}{b}^{2}{c }^{2}{x}^{4}+{P}^{4}{a}^{2}{c}^{2}{x}^{6}+4\,{P}^{4}a{b}^{3}{c}^{2}{x} ^{4}+4\,{P}^{4}ab{c}^{2}{x}^{6}+{P}^{4}{b}^{2}{c}^{2}{x}^{6}+2\,{P}^{4 }{a}^{4}{b}^{3}cx+2\,{P}^{4}{a}^{4}bc{x}^{3}+2\,{P}^{4}{a}^{3}{b}^{4}c x+4\,{P}^{4}{a}^{3}{b}^{2}c{x}^{3}+2\,{P}^{4}{a}^{3}c{x}^{5}+4\,{P}^{4 }{a}^{2}{b}^{3}c{x}^{3}+4\,{P}^{4}{a}^{2}bc{x}^{5}+2\,{P}^{4}a{b}^{4}c {x}^{3}+4\,{P}^{4}a{b}^{2}c{x}^{5}+2\,{P}^{4}ac{x}^{7}+2\,{P}^{4}{b}^{ 3}c{x}^{5}+2\,{P}^{4}bc{x}^{7}+{P}^{4}{a}^{4}{b}^{4}+2\,{P}^{4}{a}^{4} {b}^{2}{x}^{2}+{P}^{4}{a}^{4}{x}^{4}+2\,{P}^{4}{a}^{2}{b}^{4}{x}^{2}+4 \,{P}^{4}{a}^{2}{b}^{2}{x}^{4}+2\,{P}^{4}{a}^{2}{x}^{6}+{P}^{4}{b}^{4} {x}^{4}+2\,{P}^{4}{b}^{2}{x}^{6}+{P}^{4}{x}^{8}-2\,{P}^{2}{a}^{2}b{c}^ {3}{x}^{3}-2\,{P}^{2}a{b}^{2}{c}^{3}{x}^{3}-4\,{P}^{2}{a}^{3}b{c}^{2}{ x}^{2}-4\,{P}^{2}{a}^{2}{b}^{2}{c}^{2}{x}^{2}-2\,{P}^{2}{a}^{2}{c}^{2} {x}^{4}-4\,{P}^{2}a{b}^{3}{c}^{2}{x}^{2}-8\,{P}^{2}ab{c}^{2}{x}^{4}-2 \,{P}^{2}{b}^{2}{c}^{2}{x}^{4}-2\,{P}^{2}{a}^{4}bcx-4\,{P}^{2}{a}^{3}{ b}^{2}cx-4\,{P}^{2}{a}^{3}c{x}^{3}-4\,{P}^{2}{a}^{2}{b}^{3}cx-8\,{P}^{ 2}{a}^{2}bc{x}^{3}-2\,{P}^{2}a{b}^{4}cx-8\,{P}^{2}a{b}^{2}c{x}^{3}-6\, {P}^{2}ac{x}^{5}-4\,{P}^{2}{b}^{3}c{x}^{3}-6\,{P}^{2}bc{x}^{5}-2\,{P}^ {2}{a}^{4}{b}^{2}-2\,{P}^{2}{a}^{4}{x}^{2}-2\,{P}^{2}{a}^{2}{b}^{4}-8 \,{P}^{2}{a}^{2}{b}^{2}{x}^{2}-6\,{P}^{2}{a}^{2}{x}^{4}-2\,{P}^{2}{b}^ {4}{x}^{2}-6\,{P}^{2}{b}^{2}{x}^{4}-4\,{P}^{2}{x}^{6}+{a}^{2}{c}^{2}{x }^{2}-2\,ab{c}^{2}{x}^{2}+{b}^{2}{c}^{2}{x}^{2}+2\,{a}^{3}cx-2\,{a}^{2 }bcx-2\,a{b}^{2}cx+2\,{b}^{3}cx+{a}^{4}-2\,{a}^{2}{b}^{2}+{b}^{4} =0$$