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Square roots in quadratic trinomial inequalities

  1. Jun 20, 2012 #1
    How do we treat expressions under a sqaure root in inequalities ? Like for ex.

    x+4< Math.sqrt(-x^2-8x-12) (sorry, using m.physicsforums, so i dont know what to use for a root, so JAVA :p)
    I request the use of this very example.
     
  2. jcsd
  3. Jun 20, 2012 #2
    So what I like to do is ignore the inequality sign. Treat it as an = sign. Then when I have solutions for the equality I go back and test in the original equation to find the solutions to my inequality.

    So for your example.

    Square both sides.

    (x+4)^2 = -x^2 - 8x -12

    Expand

    x^2 + 8x + 16 = -x^2 - 8x - 12

    Get variables on one side and combine like terms

    2x^2 + 16x + 28 = 0

    Divide by 2

    x^2 + 8x + 14 = 0

    Solve

    x = (-8 +/- SQRT(64 - 4*1*14) ) / 2*1

    x = (-8 +/- SQRT(64 - 56 ) / 2

    x = (-8 +/- SQRT(8) / 2

    x = (-8 +/- 2*SQRT(2) / 2

    x = -4 +/- 1*SQRT(2)

    x = -4 +/- SQRT(2)

    So now that we have our two solutions we want to treat these as critical points and see what happens between them to find what our solution is.

    So I like to test points.

    I will test a point greater than both our solutions (0), a point between our two solutinos (-4) and a point less than our two solutions (-10).

    For x = 0 we get

    0+4< Math.sqrt(-0^2-8*0-12)

    4 < SQRT(-12)

    This does not solve our inequality.

    For x = -4

    -4+4< Math.sqrt(-(-4)^2-8(-4)-12)

    0 < SQRT(-16 + 32 -12)

    0 < SQRT(4)

    0 < 2

    Success!

    For x = (-10)

    (-10)+4< Math.sqrt(-(-10)^2-8(-10)-12)

    -6 < SQRT(-100 + 80 -12)

    -6 < SQRT(-32)

    This does not solve our equation.

    Now lets just double check our endpoints.

    x = -4 +/- SQRT(2)

    Remember that this solution is going to make the original equation EQUAL. That means that since we have strictly less than (<) they will not be solutions,

    So the solution to the original will be

    -4 - SQRT(2) < x < -4 + SQRT(2)

    I hope I have helped.
     
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