# Thread: Inequality question without the use of a calculator

1. ## Inequality question without the use of a calculator

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????

2. ## Re: Inequality question without the use of a calculator

Originally Posted by koolaid123
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????
Any number raised to an even power is always greater than or equal to zero.

so the smallest the function on the left side can be is 80. That number is bigger than zero.

3. ## Re: Inequality question without the use of a calculator

Originally Posted by koolaid123
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.