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Math Help - Solving Quadratic Inequalities!

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    Solving Quadratic Inequalities!

    Hello Can some one please check if i did this question right? Thank you.

    -(x-4)^2 (x-1)^2 < 0

    so, -(x-4)^2 = 0 and (x-1)^2 =0

    so, x=4 and x =1

    there fore 1<x<4

    Is this correct?
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by sakonpure6 View Post
    Hello Can some one please check if i did this question right? Thank you.
    -(x-4)^2 (x-1)^2 < 0
    so, -(x-4)^2 = 0 and (x-1)^2 =0
    so, x=4 and x =1
    there fore 1<x<4
    Is this correct?
    No it is far from correct. -(x-4)^2 (x-1)^2 is always non-positive.
    The solution set is \mathbb{R}\setminus\{1,4\}, all real numbers except 1 & 4.
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by Plato View Post
    No it is far from correct. -(x-4)^2 (x-1)^2 is always non-positive.
    The solution set is \mathbb{R}\setminus\{1,4\}, all real numbers except 1 & 4.
    The reason being is because x = 1 and x = 4 are the only values for which the expression actually equals 0, which is NOT less than 0.
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    Re: Solving Quadratic Inequalities!

    It's a quartic inequality, not quadratic. Sure it's not -(x-4)(x-1)<0?
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by Hartlw View Post
    It's a quartic inequality, not quadratic. Sure it's not -(x-4)(x-1)<0?
    A rose by any other name is still a rose.
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    Re: Solving Quadratic Inequalities!

    whoops. I missed the point. It is a quadratic:

    (x-4)^2(x-1)^2>0, so now you can take the square root to get requirement (x-4)(x-1)>0.
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by Hartlw View Post
    whoops. I missed the point. It is a quadratic:

    (x-4)^2(x-1)^2>0, so now you can take the square root to get requirement (x-4)(x-1)>0.
    Let's take a step back and think about the problem.  (x-4)^2(x-1)^2>0 is always greater than zero since everything is being squared. So the answer to this problem is ALL REAL VALUES except  x=4 and  x=1 .

    Now if the question is (x-4)^2(x-1)^2 \geq 0 , then the answer will be ALL REAL VALUES ONLY. Can you see the point here?
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by thevinh View Post
    Let's take a step back and think about the problem.  (x-4)^2(x-1)^2>0 is always greater than zero since everything is being squared. So the answer to this problem is ALL REAL VALUES except  x=4 and  x=1 .

    Now if the question is (x-4)^2(x-1)^2 \geq 0 , then the answer will be ALL REAL VALUES ONLY. Can you see the point here?
    I don't see your point. The problem starts with: -(x-4)^2 (x-1)^2 < 0
    It's pretty obvious there's a catch because when you first look at it you see a minus sign and you don't think of taking the square root. They're telling you to multiply by -1 to make it positive and then you can take the square root, which is positive by convention, and that gives the quadratic inequality which is the topic of the thread.

    As is, it is trivial.
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by Hartlw View Post
    I don't see your point. The problem starts with: -(x-4)^2 (x-1)^2 < 0
    It's pretty obvious there's a catch because when you first look at it you see a minus sign and you don't think of taking the square root. They're telling you to multiply by -1 to make it positive and then you can take the square root, which is positive by convention, and that gives the quadratic inequality which is the topic of the thread.

    As is, it is trivial.
    Like you said, first multiple both side by -1 to obtain  (x-4)^2 (x-1)^2 > 0. The point here is WE DON'T HAVE TO TAKE SQUARE ROOT, because it is obvious that anything square will always be greater than zero, except for zero itself. So the answer is ALL REAL VALUE EXCEPT 4 & 1. Beside taking a square on both side of the inequality is a dangerous maneuver. For example, would you take the square root on both side when solving this inequality:  (x+2)^2<1?

    Why do extra work to get the answer? Just simply think about the problem, analyze it, extract the most out of it.
    Last edited by thevinh; October 1st 2013 at 10:41 PM.
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by thevinh View Post
    Like you said, first multiple both side by -1 to obtain  (x-4)^2 (x-1)^2 > 0. The point here is WE DON'T HAVE TO TAKE SQUARE ROOT, because it is obvious that anything square will always be greater than zero. So the answer is ALL REAL VALUE.
    No it's not. When x = 4 or x = 1, the expression = 0, which is NOT less than 0.

    The solution is \displaystyle \begin{align*} x \in \mathbf{R} \backslash \left\{ 1, 4 \right\} \end{align*}.
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by Prove It View Post
    No it's not. When x = 4 or x = 1, the expression = 0, which is NOT less than 0.

    The solution is \displaystyle \begin{align*} x \in \mathbf{R} \backslash \left\{ 1, 4 \right\} \end{align*}.
    Sorry, I meant ALL REAL VALUE EXCEPT {1,4}. Thanks.
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    Re: Solving Quadratic Inequalities!

    thevinh. You didn't finish the inequality. Assume:

    0<(x+2)^2<1
    0< x+2 <1
    -2 < x < -1
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    Re: Solving Quadratic Inequalities!

    a^2>0 is either trivial (a='0), or a>0. Topic of thread tells you which.

    Look at it another way:
    Let 0<a^2<N, then 0<a<sqrtN.
    Let N -> infinity, then 0<a.
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by Hartlw View Post
    thevinh. You didn't finish the inequality. Assume:

    0<(x+2)^2<1
    0< x+2 <1
    -2 < x < -1
    Hartlw. For the example  (x+2)^2<1, would it still be true if x=-2.5? The right answer is  -3<x<-1. The point is that it is a fatal mistake to take square root both sides when solving an inequality. The best way to solve this problem is to expand them, bring everything to one side, factor and solve. See below.

     (x+2)^2<1~ \Leftrightarrow~ x^2+4x+4<1~ \Leftrightarrow~ x^2+4x+3<0~ \Leftrightarrow~ (x+1)(x+3)<0

    This is a proper way of solving inequality that ensures no lost of solution.

    From your solution, when x=-2 the inequality still holds.
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    Re: Solving Quadratic Inequalities!

    Quote Originally Posted by thevinh View Post
    Hartlw. For the example  (x+2)^2<1, would it still be true if x=-2.5? The right answer is  -3<x<-1. The point is that it is a fatal mistake to take square root both sides when solving an inequality. The best way to solve this problem is to expand them, bring everything to one side, factor and solve. See below.
     (x+2)^2<1~ \Leftrightarrow~ x^2+4x+4<1~ \Leftrightarrow~ x^2+4x+3<0~ \Leftrightarrow~ (x+1)(x+3)<0
    This is a proper way of solving inequality that ensures no lost of solution.[/B]
    I disagree with there being one proper way of solving anything.

    For example: {a^2} \le {b^2} \leftrightarrow |a| \le |b|.

    So in this case (x+2)^2<1  \leftrightarrow |x+2|< 1 or -3<x<-1.
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