Question

If the ratio of the roots of x2+bx+c=0; x2+qx+r=0 are the same them

Answer 1

Completing Question : Find relationship between b, c, q,r.

Final Answer :
${q}^{2} c = r {b}^{2}$

Steps and Understanding :
1) We know that,
absolute value of difference between roots of
$a {x}^{2} + bx + c = 0$

=
$= \frac{ \sqrt{ {b}^{2} - 4ac} }{ |a| }$

2) Now, we have
${x}^{2} + bx + c = 0 \\ {x}^{2} + qx + r = 0$
Let the roots be respectively,
$\alpha (1) \: . \beta (1). \: \: \: \: \alpha (2). \: \beta (2)$

3) We have,
$\frac{ \alpha (1)}{ \beta (1)} = \frac{ \alpha (2)}{ \beta (2)}$
where α(1) > β(1) and α(2)> β(2) .

By Componendo and Dividendo,
$\frac{ \alpha (1) + \beta(1) }{ \alpha (1) - \beta (1)} = \frac{ \alpha (2) + \beta (2)}{ \alpha (2) - \beta (2)} \\ = > \frac{ - b}{ \sqrt{ {b}^{2} - 4c} } = \frac{ - q}{ \sqrt{ {q}^{2} - 4r } } \\ = > \frac{ {b}^{2} - 4c }{ {b}^{2} } = \frac{ {q}^{2} - 4r}{ {q}^{2} } \\ = > \frac{ - 4c}{ {b}^{2} } = \frac{ - 4r}{ {q}^{2} } \\ = > {q}^{2} c = r {b}^{2}$

This is the required Relationship.
$\boxed { {q}^{2}c = r {b}^{2} }$