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Thread: [ASK] Proof of Some Quadratic Functions

  1. #1
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    [ASK] Proof of Some Quadratic Functions

    So, I found these statements and I need your assistance to prove them since my body condition is not fit enough to think that much.
    1. The quadratic equation whose roots are k less than the roots of $\displaystyle ax^2+bx+c=0$ is $\displaystyle a(x+k)^2+b(x+k)+c=0$.
    2. The quadratic equation whose roots are k more than the roots of $\displaystyle ax^2+bx+c=0$ is $\displaystyle a(x-k)^2+b(x-k)+c=0$.
    3. The quadratic equation whose roots are n times the roots of $\displaystyle ax^2+bx+c=0$ is $\displaystyle ax+bnx+cn^2=0$.
    4. The quadratic equation whose roots are negations of the roots of $\displaystyle ax^2+bx+c=0$ is $\displaystyle ax^2-bx+c=0$.
    5. The quadratic equation whose roots are inverses of the roots of $\displaystyle ax^2+bx+c=0$ is $\displaystyle cx^2+bx+a=0$.
    6. The quadratic equation whose roots are squareroots of the roots of $\displaystyle ax^2+bx+c=0$ is $\displaystyle a^2x^2-(b^2-2ac)+c^2=0$.
    Thanks for your help.
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    Re: [ASK] Proof of Some Quadratic Functions

    1) and 2), set (for example) $y=x+k$, the roots of the quadratic in $y$ are those of the roots of the first quadratic in the question. Replace $y$ with the expression for $x$ and you have your result.
    3) is similar: $y = nx$. 4) is a special case of 3).
    In 5) "inverse" presumably means "reciprocal". Set $y = \frac1x$ and form a quadratic from the resultant equation.
    6) requires something mildly different, but it ought to be pretty clear how to approach it from what I've already written.
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  3. #3
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    Re: [ASK] Proof of Some Quadratic Functions

    Mr.Fly, one is innocent until proven guilty.
    Soooo....those 6 statements are correct until proven incorrect...
    Soooo....relax
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    Re: [ASK] Proof of Some Quadratic Functions

    (6)
    if we take $a=1,b=-5,c=4$ then the roots of

    $a^2 x^2-(b^2-2ac)x +c^2=x^2-17x+16=0$

    are the squares (not the square roots) of the roots of

    $ax^2+bx+c=x^2-5x+4=0$
    Last edited by Idea; Apr 8th 2019 at 07:21 AM.
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    Re: [ASK] Proof of Some Quadratic Functions

    Quote Originally Posted by Archie View Post
    6) requires something mildly different, but it ought to be pretty clear how to approach it from what I've already written.
    Actually, I think it's a lot different. More along the lines of identifying that for roots $r_1,r_2$ of the first equation we have $\frac{c}{a} = r_1r_2$ and $-\frac{b}{a}=r_1+r_2$. For the equation $Ax^2+Bx +C=0$ with roots $r_1^2, r_2^2$ we therefore have $\frac{C}{A} = r_1^2 r_2^2 = \frac{c^2}{a^2}$ and $-\frac{B}{A}=r_1^2+r_2^2 = (r_1+r_2)^2 - 2r_1r_2$.
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    Re: [ASK] Proof of Some Quadratic Functions

    Quote Originally Posted by Archie View Post
    1) and 2), set (for example) $y=x+k$, the roots of the quadratic in $y$ are those of the roots of the first quadratic in the question. Replace $y$ with the expression for $x$ and you have your result.
    3) is similar: $y = nx$. 4) is a special case of 3).
    In 5) "inverse" presumably means "reciprocal". Set $y = \frac1x$ and form a quadratic from the resultant equation.
    6) requires something mildly different, but it ought to be pretty clear how to approach it from what I've already written.
    Yes. "Inverse" does mean "reciprocal".
    For number 6, I mistook "whose roots are squares" as "squareroots".

    Quote Originally Posted by DenisB View Post
    Mr.Fly, one is innocent until proven guilty.
    Soooo....those 6 statements are correct until proven incorrect...
    Soooo....relax
    Don't worry, Denis McField, someone at MIF has already proven their innocance. Now they can Rest in Peace (Wait... Doesn't that mean they're executed?).

    Quote Originally Posted by Idea View Post
    (6)
    if we take $a=1,b=-5,c=4$ then the roots of

    $a^2 x^2-(b^2-2ac)x +c^2=x^2-17x+16=0$

    are the squares (not the square roots) of the roots of

    $ax^2+bx+c=x^2-5x+4=0$
    Thank you. I did mistook "whose roots are squares" as "squareroots".

    Quote Originally Posted by Archie View Post
    Actually, I think it's a lot different. More along the lines of identifying that for roots $r_1,r_2$ of the first equation we have $\frac{c}{a} = r_1r_2$ and $-\frac{b}{a}=r_1+r_2$. For the equation $Ax^2+Bx +C=0$ with roots $r_1^2, r_2^2$ we therefore have $\frac{C}{A} = r_1^2 r_2^2 = \frac{c^2}{a^2}$ and $-\frac{B}{A}=r_1^2+r_2^2 = (r_1+r_2)^2 - 2r_1r_2$.
    I mistook "whose roots are squares" as "squareroots".
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