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On a math test, one of the questions was to solve [itex]-\sqrt{7-x}=-\frac{x^2}{2}+12x-10[/itex]. I solved graphically with a calculator, but later tried to solve algebraically, when I had more time. The equation is equivalent (with extraneous solutions) to [itex]x^4 + 48x^3 +536x^2 -956x + 372=0[/itex]. This quartic has no rational roots (the rational roots theorem gives ±1, ±2, ±3, ±4, ±6, ±12, ±31, ±62, ±93, ±124, ±186, ±372, but none of these are zeros. I have shown that all real zeros lie in (-62, 2), but this is as far as I have gotten. What is the next step after the Rational Roots Theorem (RRT) has failed to find a root?
Thanks!
Thanks!