SOLUTION: x^4+100>29x^2 That greater than sign should be a greater than or equal to sign, but I didn't know how to make it on here.

Question 171339This question is from textbook
: x^4+100>29x^2
That greater than sign should be a greater than or equal to sign, but I didn't know how to make it on here. This question is from textbook

Found 2 solutions by stanbon, gonzo:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
x^4+100>29x^2
That greater than sign should be a greater than or equal to sign,
--------------------
1st: Solve the EQUALITY
x^4 -29x^2 + 100 = 0
(x^2-25)(x^2-4) = 0
x = 5 or -5 or 2 or -2
------------------------------
Draw a number line and plot -5, -2, 2, 5
-----------------------------------------
2nd: Check a test value in each interval to see where the INEQUALITY solutions are
Test x = -10: (100-25)(100-4)> 0; true so solutions in (-inf,-5)
Test x = -3: (9-25)(9-4)> 0 ; false so no solutions in that interval
Test x = 0: (0-25)(0-4)> 0 ; true so solutions in (-2,2)
Test x = 3: (9-25)(9-4) >0 ; false so no solutions in that interval
Test x = 10: (100-25)(100-4)>0: true so solutions in (5,+inf)
=========================================
Final Solution: (-inf,-5]U[-2,2]U[5,+inf)
=========================================
Cheers,
Stan H.
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
x^4 + 100 >= 29x^2
subtract 29x^2 from both sides:
x^4 - 29x^2 + 100 >= 0
let y = x^2
your equation becomes:
y^2 - 29y + 100 >= 0
this looks like a quadratic equation.
solve it that way.
you get:
(y-25) * (y-4) >= 0
this means:
y >= 25
y >= 4
-----
since y = x^2, then:
x^2 >= 25
or
x^2 >= 4
this means that:
x >= +/- 5
or
x >= +/- 2
-----
your original equation is:
x^4 + 100 >= 29x^2
substitute x = 5 into it.
(5)^4 + 100 = 29*(5)^2
625 + 100 = 29*25
725 = 725
equation is true.
x = 5 satisfies the equation.
take any value of x > 5.
try 6:
x^4 + 100 >= 29x^2
(6)^4 + 100 > 29*(6)^2
1396 > 1044
equation is true.
x >= 5 is a good answer.
since x has to be >= 5, then x >= -5 will not be good.
i tried it and it isn't.
x = -5 is good, but x > -5 is not good.
-----
likewise, i doubt x >= 2 is going to work either.
try x = 4
x^4 + 100 >= 29x^2 becomes:
(4)^4 + 100 >= 29*(4)^2
356 >= 464
equation is false
x = 4 is no good
x >= 2 will not be a good answer.
x >= -2 will not be a good answer either.
i did them anyway.
-----
try x = 2
x^4 + 100 >= 29x^2
(2)^4 + 100 = 29*(2)^2
116 = 116
equation is true
x = 2 is good.
try x = 3 (greater than 2)
(3)^4 + 100 > 29*(3)^2
181 > 261
equation is false.
x > 2 is not good
so x >= 2 is not good since the > part doesn't work.
-----
x >= -2 didn't work either.
-----
the only good answer is:
x >= 5
-----
the key to solving this was recognizing that x^4 - x^2 + a constant was a quadratic equation in disguise.
by substituting y = x^2 you forced it to be a quadratic and could solve it that way.
the other kicker was the >= sign.
it's tricky and you have to make sure you satisfy all requirements.
satisfying = was not enough.
you have to satisfy > also.
-----

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