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Thread: The quadratic Problem

  1. #1
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    The quadratic Problem

    I am not able to solve this problem

    if $\displaystyle a^2+b^2=1$ then find the range of $\displaystyle a+b$

    what I am doing is
    $\displaystyle (a+b)^2>0$
    $\displaystyle a^2+b^2+2ab>0$ where $\displaystyle a^2+b^2=1$
    $\displaystyle 1+2ab>0$
    $\displaystyle 2ab>-1$

    Case 2
    $\displaystyle (a-b)^2>0$
    $\displaystyle a^2+b^2-2ab>0$ where $\displaystyle a^2+b^2=1$
    $\displaystyle 1-2ab>0$
    $\displaystyle 2ab<1$

    $\displaystyle -1<2ab<1$

    $\displaystyle a+b=\sqrt{a^2+b^2+2ab}$
    $\displaystyle a+b=\sqrt{1+2ab}$
    $\displaystyle a+b=\sqrt{1-1}$ $\displaystyle -1<2ab<1$ to find minimum
    $\displaystyle a+b=0$


    $\displaystyle a+b=\sqrt{1+1}$ $\displaystyle -1<2ab<1$ to find maximum
    $\displaystyle a+b=\sqrt{2}$


    The range I am getting is $\displaystyle (0,\sqrt{2})$ but the answer is ($\displaystyle (-\sqrt{2},\sqrt{2})$
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  2. #2
    MHF Contributor

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    Re: The quadratic Problem

    I would do this a completely different way. If we use x and y rather than a and b (it is still exactly the same problem, of course), we can think of "a^2+ b^2= 1" (x^2+ y^2= 1) as the equation of a circle with center at (0, 0) and radius 1. "a+ b= c" or x+ y= c, for c a fixed number, is a line sloping down to the left at a 45 degree angle. The smallest and largest possible values of c are when that line is tangent to the circle which happen at the endpoints of the diameter perpendicular to the line: that is the line y= x which crosses the circle $\displaystyle x^2+ y^2= 1$ when $\displaystyle x^2+ x^2= 2x^2= 1$ so that $\displaystyle x= y= \pm\sqrt{2}/2$. Taking $\displaystyle x= a= b= \sqrt{2}/2$, $\displaystyle a+ b= \sqrt{2}/2+\sqrt{2}/2= \sqrt{2}$. But taking $\displaystyle x= a= b= -\sqrt{2}/2$ we get $\displaystyle a+ b= -\sqrt{2}$.

    Perhaps you forgot the $\displaystyle \pm$ on the square root?
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