Solve the equation: $$x^4+97x^2+1296=0$$
Explanation
Substitute $$y = x^2$$: $$y^2+97y+1296=0$$ Find discriminant of quadratic equation: $$D=(97)^2-4\cdot1\cdot1296=9409-5184=4225$$ Discriminant is greater than zero, so there are two (unequal) real solutions of the equation: $$y_{1,2}=\dfrac{-97\pm65}{2\cdot1}={-81,-16}$$ There two quadratic equations: $$(1) x^2=-81$$ $$(2) x^2=-16$$ Quadratic equations (1) and (2) have no real solutions. $$x \in \emptyset$$
Theory
• Solution of quadratic equations. A quadratic equation is any equation having the form $$ax^2+bx+c=0$$ where $$x$$ represents an unknown, and $$a,b$$ and $$c$$ represent known numbers such that $$a$$ is not equal to 0. There are two stages in solving quadratic equations. 1) Find discriminant using formula $$D=b^2-4ac$$
2) Check the sign of discriminant: 2.1) If $$D>0$$, then equation has two (unequal) real solutions $$x_{1,2}=\dfrac{-b\pm\sqrt{D}}{2\cdot a}$$
2.2) If $$D=0$$, then equation has two (equal) real solutions $$x=\dfrac{-b}{2\cdot a}$$
2.3) If $$D<0$$, then equation has no real solutions.
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