1. ## Comparing coefficients in quadratic?

p(x^2)+qx=r=0 and d(x^2)+ex+f=0 with one common root

can i write (p/d)=(q/e)=(r/f) ?

2. ## Re: Comparing coefficients in quadratic?

Originally Posted by AaPa

p(x^2)+qx=r=0 and d(x^2)+ex+f=0 with one common root

can i write (p/d)=(q/e)=(r/f) ?
Can you?

Compare: $2x^2 + x - 1$ vs. $2x^2 + 3x - 2$

-Dan

3. ## Re: Comparing coefficients in quadratic?

no.
so when can we compare coefficients?

4. ## Re: Comparing coefficients in quadratic?

If $p(x),q(x)$ are two polynomials with the exact same set of roots and the same degree, then $p(x) = kq(x)$ where $k$ is some constant. That is when you can compare coefficients.

5. ## Re: Comparing coefficients in quadratic?

All my confusions which troubled me have disappeared. You really made me think on my own.

Thank you very very muchh!!

6. ## Re: Comparing coefficients in quadratic?

hey

like comparing coefficients.

can we compare exponents as in this given equation

[(a^n) + (b^n)]/[(a^n-1) + (b^n-1)] = (a+b)/2
(n-1 is exponent)

to get n=1

if yes then what is the condition that allows us to compare?

7. ## Re: Comparing coefficients in quadratic?

$a^n + b^n / a^n-1 +b^n-1 = a+b / 2$

8. ## Re: Comparing coefficients in quadratic?

Maybe this will help

$x^2 + b x + c$

$x^2 + s x + t$

have a root in common if and only if

$c^2 - b c s + c s^2 + b^2 t - 2 c t - b s t + t^2 = 0$

9. ## Re: Comparing coefficients in quadratic?

$\frac {a^n + b^n} { a^{n-1} + b^{n-1}} = \frac{a+b} {2}$

by comparing exponents we get n=1
exactly when can we compare exponents like this?

10. ## Re: Comparing coefficients in quadratic?

Originally Posted by AaPa
$\frac {a^n + b^n} { a^{n-1} + b^{n-1}} = \frac{a+b} {2}$

by comparing exponents we get n=1
exactly when can we compare exponents like this?
Well...We can solve this in terms of a and b: b = -a for odd n, so there is no specific relationship needed for n. But generally we can match coefficients of terms on each side of an equation if the terms are members of an orthogonal series. The "power series" is such a series: $a_0 + a_1x + a_2 x^2 + ...$. Many other orthogonal expansions exist but you'll see the power series the most often.

-Dan

11. ## Re: Comparing coefficients in quadratic?

let us say that a and b are positive distinct real natural numbers....is there any other value of n that satisfies the equation? how do we tell?