[ASK] Proof of Some Quadratic Functions

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SUMMARY

The discussion focuses on proving various transformations of quadratic equations based on their roots. Specifically, it addresses how to derive new quadratic equations from the original equation $$ax^2 + bx + c = 0$$ when the roots are modified by addition, subtraction, multiplication, negation, inversion, and squaring. Key transformations include: for roots decreased by k, the equation is $$a(x+k)^2 + b(x+k) + c = 0$$; for roots increased by k, it is $$a(x-k)^2 + b(x-k) + c = 0$$; and for roots negated, the equation becomes $$ax^2 - bx + c = 0$$. Additionally, a correction is noted regarding the squaring of roots, which should read as $$a^2x^2 - (b^2 - 2ac)x + c^2 = 0$$ instead of squareroots.

PREREQUISITES
  • Understanding of quadratic equations and their standard form.
  • Familiarity with polynomial transformations and root modifications.
  • Knowledge of Vieta's formulas and their application in polynomial equations.
  • Basic algebraic manipulation skills, including factoring and expanding expressions.
NEXT STEPS
  • Study the derivation of quadratic equations from root transformations using Vieta's formulas.
  • Explore the implications of polynomial transformations in higher-degree polynomials.
  • Learn about the graphical interpretations of root modifications in quadratic functions.
  • Investigate the properties of quadratic equations under various algebraic operations.
USEFUL FOR

Mathematicians, educators, and students studying algebra, particularly those focusing on quadratic equations and their properties. This discussion is beneficial for anyone looking to deepen their understanding of polynomial transformations and their applications.

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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