Explain without a calculator why the inequality has no solution????
i noe how to do this with a calcl but i dont noe how to explain it....plzzz help????
$\displaystyle \displaystyle \begin{align*} 2x^{24} + x^4 + 15x^2 + 80 &= 2x^{24} + x^4 + 15x^2 + \left(\frac{15}{2}\right)^2 - \left(\frac{15}{2}\right)^2 + 80 \\ &= 2x^{24} + \left( x^2 + \frac{15}{2} \right)^2 - \frac{225}{4} + \frac{320}{4} \\ &= 2x^{24} + \left( x^2 + \frac{15}{2} \right)^2 + \frac{95}{4} \end{align*}$
Each term is always nonnegative, so their sum is always nonnegative.
Edit: of course, completing the square in this case proves to be pointless, as your function is already the sum of even-powers which always yield nonnegative results, as the post above me points out.