Inequality question without the use of a calculator

• Oct 4th 2012, 07:25 PM
koolaid123
Inequality question without the use of a calculator
Explain without a calculator why the inequality http://www.mathway.com/math_image.as...020?p=172?p=22 has no solution????

i noe how to do this with a calcl but i dont noe how to explain it....plzzz help????
• Oct 4th 2012, 08:05 PM
TheEmptySet
Re: Inequality question without the use of a calculator
Quote:

Originally Posted by koolaid123
Explain without a calculator why the inequality http://www.mathway.com/math_image.as...020?p=172?p=22 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.
• Oct 4th 2012, 08:56 PM
Prove It
Re: Inequality question without the use of a calculator
Quote:

Originally Posted by koolaid123
Explain without a calculator why the inequality http://www.mathway.com/math_image.as...020?p=172?p=22 has no solution????

i noe how to do this with a calcl but i dont noe how to explain it....plzzz help????

\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.