[ASK] Proof of Some Quadratic Functions

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Discussion Overview

The discussion revolves around proving various statements regarding quadratic functions and their transformations. Participants explore how modifications to the roots of a given quadratic equation, $$ax^2+bx+c=0$$, lead to new quadratic equations. The scope includes theoretical reasoning and mathematical proofs.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents several statements about quadratic equations and requests assistance in proving them, including transformations such as shifting roots by a constant, scaling roots, negating roots, and taking inverses.
  • Another participant suggests using the quadratic formula to verify the first statement, showing the roots of the transformed equation and questioning the validity of the conjecture.
  • Some participants express confusion about the mathematical steps involved, particularly in deriving the differences between roots.
  • One participant proposes an alternative approach by considering the roots of the transformed equation and how they relate to the original equation, arguing against the necessity of the quadratic formula.
  • A later reply introduces a general lemma about polynomial equations and transformations, applying it to the first five statements while correcting the sixth statement regarding the roots of squares versus square roots.
  • Participants engage in clarifying the correct interpretation of the sixth statement, with one acknowledging a misunderstanding between "squareroots" and "squares" of the roots.

Areas of Agreement / Disagreement

There is no consensus on the correctness of the original statements, particularly the sixth one, which is identified as incorrect by one participant. Multiple viewpoints exist regarding the best approach to proving the statements, with some favoring the quadratic formula and others preferring a more conceptual method.

Contextual Notes

Participants express uncertainty about specific mathematical steps and the implications of different transformations on the roots of the quadratic equations. The discussion reflects varying levels of understanding and approaches to the problem.

Monoxdifly
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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 $$ax^2+bx+c=0$$ is $$a(x+k)^2+b(x+k)+c=0$$.
2. The quadratic equation whose roots are k more than the roots of $$ax^2+bx+c=0$$ is $$a(x-k)^2+b(x-k)+c=0$$.
3. The quadratic equation whose roots are n times the roots of $$ax^2+bx+c=0$$ is $$ax+bnx+cn^2=0$$.
4. The quadratic equation whose roots are negations of the roots of $$ax^2+bx+c=0$$ is $$ax^2-bx+c=0$$.
5. The quadratic equation whose roots are inverses of the roots of $$ax^2+bx+c=0$$ is $$cx^2+bx+a=0$$.
6. The quadratic equation whose roots are squareroots of the roots of $$ax^2+bx+c=0$$ is $$a^2x^2-(b^2-2ac)+c^2=0$$.
Thanks for your help.
 
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Monoxdifly said:
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 $$ax^2+bx+c=0$$ is $$a(x+k)^2+b(x+k)+c=0$$.
2. The quadratic equation whose roots are k more than the roots of $$ax^2+bx+c=0$$ is $$a(x-k)^2+b(x-k)+c=0$$.
3. The quadratic equation whose roots are n times the roots of $$ax^2+bx+c=0$$ is $$ax+bnx+cn^2=0$$.
4. The quadratic equation whose roots are negations of the roots of $$ax^2+bx+c=0$$ is $$ax^2-bx+c=0$$.
5. The quadratic equation whose roots are inverses of the roots of $$ax^2+bx+c=0$$ is $$cx^2+bx+a=0$$.
6. The quadratic equation whose roots are squareroots of the roots of $$ax^2+bx+c=0$$ is $$a^2x^2-(b^2-2ac)+c^2=0$$.
Thanks for your help.
Have you solved the quadratics for these to see? I'm using the "brute force" approach here. (If you would like to be more sophisticated you can use the Vieta formulas, but I don't think you gain any advantage using them.) If you like you can use the concept of translations and dilations of the graphs but I feel that working with the quadratic formula is the best bet for understanding the "why" of it. Your method of choice will be what unit you are currently studying.

1)
[math]ax^2 + bx + c = 0[/math] has roots [math]x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/math]

[math]a(x + k)^2 + b(x + k) + c = 0 \implies ax^2 + (2ak + b)x + (k^2 + bk + c) = 0[/math]

has roots [math]x = \dfrac{-(2ak + b) \pm \sqrt{ (2ak + b)^2 - 4a(k^2 + bk + c) }}{2a}[/math]

This simplifies to [math]x = \dfrac{-(2ak + b) \pm \sqrt{b^2 - 4ac}}{2a}[/math]

So is the conjecture true?

The rest are of a similar nature.

-Dan
 
I was stuck at exactly the line you ended in. And... I still don'e get the next line...
 
Monoxdifly said:
I was stuck at exactly the line you ended in. And... I still don'e get the next line...
What is the difference between the roots?
[math]\dfrac{-(2ak + b) + \sqrt{b^2 - 4ac}}{2a} - \dfrac{-b + \sqrt{b^2 - 4ac}}{2a} = [/math]?

-Dan
 
I don't see any reason to appeal to the quadratic formula. For the first one, suppose that u and v are roots to the equation $a(x+ k)^2+ b(x+ k)+ c= 0$. Then p= u+ k and q= v+ k satisfy $ax^2+ bx+ c= 0$.
 
In general:

If $p(x)=0$ is a polynomial equation of degree $n$ with roots $\alpha_1,\ldots,\alpha_n$, and $f:\mathbb R\to\mathbb R$ is an invertible function, then the polynomial equation whose roots are $f(\alpha_1),\ldots,f(\alpha_n)$ is
$$p\left(f^{-1}(x)\right)\ =\ 0.$$
Proof:

For each $i=1,\ldots,n$,
$$p\left(f^{-1}(f(\alpha_i))\right)\ =\ p(\alpha_i)\ =\ 0$$
as $\alpha_i$ is a root of $p(x)=0$.

Thus $\alpha$ is a root of $p(x)=0$ if and only if $f(\alpha)$ is a root of $p\circ f^{-1}(x)=0$.

This lemma applies to statements 1–5, where $p(x)=ax^2+bx+c$ and

  1. $f(x)=x-k$,
  2. $f(x)=x+k$,
  3. $f(x)=nx$,
  4. $f(x)=-x$,
  5. $f(x)=\dfrac1x$.

Statement 6 is incorrect: it should read

Monoxdifly said:
6. The quadratic equation whose roots are squares of the roots of $$ax^2+bx+c=0$$ is $$a^2x^2-(b^2-2ac){\color{red}x}+c^2=0$$.
 
topsquark said:
What is the difference between the roots?
[math]\dfrac{-(2ak + b) + \sqrt{b^2 - 4ac}}{2a} - \dfrac{-b + \sqrt{b^2 - 4ac}}{2a} = [/math]?
–k?

Country Boy said:
I don't see any reason to appeal to the quadratic formula. For the first one, suppose that u and v are roots to the equation $a(x+ k)^2+ b(x+ k)+ c= 0$. Then p= u+ k and q= v+ k satisfy $ax^2+ bx+ c= 0$.
Gotta take notes to try using that someday.

Olinguito said:
In general:

If $p(x)=0$ is a polynomial equation of degree $n$ with roots $\alpha_1,\ldots,\alpha_n$, and $f:\mathbb R\to\mathbb R$ is an invertible function, then the polynomial equation whose roots are $f(\alpha_1),\ldots,f(\alpha_n)$ is
$$p\left(f^{-1}(x)\right)\ =\ 0.$$
Proof:

For each $i=1,\ldots,n$,
$$p\left(f^{-1}(f(\alpha_i))\right)\ =\ p(\alpha_i)\ =\ 0$$
as $\alpha_i$ is a root of $p(x)=0$.

Thus $\alpha$ is a root of $p(x)=0$ if and only if $f(\alpha)$ is a root of $p\circ f^{-1}(x)=0$.

This lemma applies to statements 1–5, where $p(x)=ax^2+bx+c$ and

  1. $f(x)=x-k$,
  2. $f(x)=x+k$,
  3. $f(x)=nx$,
  4. $f(x)=-x$,
  5. $f(x)=\dfrac1x$.

Statement 6 is incorrect: it should read

Thank you. I did mistook "whose roots are squares" as "squareroots".
 

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