# Thread: Show that the equation has real and distinct roots

1. ## Show that the equation has real and distinct roots

Question: show that the roots of the equation $\displaystyle (px-1)^2 +3px -5 = 0$ are real and distinct for all real values of $\displaystyle p$ and $\displaystyle p \neq 0$

My workings:
$\displaystyle (px-1)^2 +3px -5 = 0$

$\displaystyle p^2x^2 -2px +1 +3px -5 = 0$

$\displaystyle p^2x^2 +px -4= 0$

Here $\displaystyle a = p^2, b = p, c = -4$. Now

$\displaystyle b^2 -4ac = p^2 -4p^2(-4)$

$\displaystyle = p^2 + 16p^2$

$\displaystyle = 17p^2$

2. Originally Posted by PythagorasNeophyte
Question: show that the roots of the equation $\displaystyle (px-1)^2 +3px -5 = 0$ are real and distinct for all real values of $\displaystyle p$ and $\displaystyle p \neq 0$

My workings:
$\displaystyle (px-1)^2 +3px -5 = 0$

$\displaystyle p^2x^2 -2px +1 +3px -5 = 0$

$\displaystyle p^2x^2 +px -4= 0$

Here $\displaystyle a = p^2, b = p, c = -4$. Now

$\displaystyle b^2 -4ac = p^2 -4p^2(-4)$

$\displaystyle = p^2 + 16p^2$

$\displaystyle = 17p^2$

A square is allways positive or zero. If you multiply a square by a positive number the result is positive too.

You get two different values if the discriminant unequals zero. Therefore you'll get real and distinct roots of the equation if $\displaystyle 17p^2\ne 0 ~\implies~p\ne 0$

3. Originally Posted by PythagorasNeophyte
Question: show that the roots of the equation $\displaystyle (px-1)^2 +3px -5 = 0$ are real and distinct for all real values of $\displaystyle p$ and $\displaystyle p \neq 0$

My workings:
$\displaystyle (px-1)^2 +3px -5 = 0$

$\displaystyle p^2x^2 -2px +1 +3px -5 = 0$

$\displaystyle p^2x^2 +px -4= 0$
Suppose $\displaystyle $$p>0 , then by Descartes rule of signs there is exactly one positive roots, and exactly one negative root. Suppose \displaystyle$$p<0$, then by Descartes rule of signs there is exactly one positive root and one negative root.

QED

CB

4. Originally Posted by PythagorasNeophyte
Question: show that the roots of the equation $\displaystyle (px-1)^2 +3px -5 = 0$ are real and distinct for all real values of $\displaystyle p$ and $\displaystyle p \neq 0$

My workings:
$\displaystyle (px-1)^2 +3px -5 = 0$

$\displaystyle p^2x^2 -2px +1 +3px -5 = 0$

$\displaystyle p^2x^2 +px -4= 0$

Here $\displaystyle a = p^2, b = p, c = -4$. Now

$\displaystyle b^2 -4ac = p^2 -4p^2(-4)$

$\displaystyle = p^2 + 16p^2$

$\displaystyle = 17p^2$

Do you recall why you calculated that? It is the discriminant of the equation and a quadratic equation has distinct real roots if and only if its discriminant is positive. Since $\displaystyle p\ne 0$ is a condition of the problem, what can you say about $\displaystyle p^2$?