document.write( "Question 171339This question is from textbook
\n" ); document.write( ": x^4+100>29x^2\r
\n" ); document.write( "\n" ); document.write( "That greater than sign should be a greater than or equal to sign, but I didn't know how to make it on here. \n" ); document.write( "
Algebra.Com's Answer #126504 by gonzo(654)\"\" \"About 
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x^4 + 100 >= 29x^2
\n" ); document.write( "subtract 29x^2 from both sides:
\n" ); document.write( "x^4 - 29x^2 + 100 >= 0
\n" ); document.write( "let y = x^2
\n" ); document.write( "your equation becomes:
\n" ); document.write( "y^2 - 29y + 100 >= 0
\n" ); document.write( "this looks like a quadratic equation.
\n" ); document.write( "solve it that way.
\n" ); document.write( "you get:
\n" ); document.write( "(y-25) * (y-4) >= 0
\n" ); document.write( "this means:
\n" ); document.write( "y >= 25
\n" ); document.write( "y >= 4
\n" ); document.write( "-----
\n" ); document.write( "since y = x^2, then:
\n" ); document.write( "x^2 >= 25
\n" ); document.write( "or
\n" ); document.write( "x^2 >= 4
\n" ); document.write( "this means that:
\n" ); document.write( "x >= +/- 5
\n" ); document.write( "or
\n" ); document.write( "x >= +/- 2
\n" ); document.write( "-----
\n" ); document.write( "your original equation is:
\n" ); document.write( "x^4 + 100 >= 29x^2
\n" ); document.write( "substitute x = 5 into it.
\n" ); document.write( "(5)^4 + 100 = 29*(5)^2
\n" ); document.write( "625 + 100 = 29*25
\n" ); document.write( "725 = 725
\n" ); document.write( "equation is true.
\n" ); document.write( "x = 5 satisfies the equation.
\n" ); document.write( "take any value of x > 5.
\n" ); document.write( "try 6:
\n" ); document.write( "x^4 + 100 >= 29x^2
\n" ); document.write( "(6)^4 + 100 > 29*(6)^2
\n" ); document.write( "1396 > 1044
\n" ); document.write( "equation is true.
\n" ); document.write( "x >= 5 is a good answer.
\n" ); document.write( "since x has to be >= 5, then x >= -5 will not be good.
\n" ); document.write( "i tried it and it isn't.
\n" ); document.write( "x = -5 is good, but x > -5 is not good.
\n" ); document.write( "-----
\n" ); document.write( "likewise, i doubt x >= 2 is going to work either.
\n" ); document.write( "try x = 4
\n" ); document.write( "x^4 + 100 >= 29x^2 becomes:
\n" ); document.write( "(4)^4 + 100 >= 29*(4)^2
\n" ); document.write( "356 >= 464
\n" ); document.write( "equation is false
\n" ); document.write( "x = 4 is no good
\n" ); document.write( "x >= 2 will not be a good answer.
\n" ); document.write( "x >= -2 will not be a good answer either.
\n" ); document.write( "i did them anyway.
\n" ); document.write( "-----
\n" ); document.write( "try x = 2
\n" ); document.write( "x^4 + 100 >= 29x^2
\n" ); document.write( "(2)^4 + 100 = 29*(2)^2
\n" ); document.write( "116 = 116
\n" ); document.write( "equation is true
\n" ); document.write( "x = 2 is good.
\n" ); document.write( "try x = 3 (greater than 2)
\n" ); document.write( "(3)^4 + 100 > 29*(3)^2
\n" ); document.write( "181 > 261
\n" ); document.write( "equation is false.
\n" ); document.write( "x > 2 is not good
\n" ); document.write( "so x >= 2 is not good since the > part doesn't work.
\n" ); document.write( "-----
\n" ); document.write( "x >= -2 didn't work either.
\n" ); document.write( "-----
\n" ); document.write( "the only good answer is:
\n" ); document.write( "x >= 5
\n" ); document.write( "-----
\n" ); document.write( "the key to solving this was recognizing that x^4 - x^2 + a constant was a quadratic equation in disguise.
\n" ); document.write( "by substituting y = x^2 you forced it to be a quadratic and could solve it that way.
\n" ); document.write( "the other kicker was the >= sign.
\n" ); document.write( "it's tricky and you have to make sure you satisfy all requirements.
\n" ); document.write( "satisfying = was not enough.
\n" ); document.write( "you have to satisfy > also.
\n" ); document.write( "-----\r
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