Question

the equations x³ + 5x² + px + q = 0 and x³ + 7x² + px + r = 0, (p, q, r E R) have exactly two roots common, then p:q:r is​

Answer 1

Ordered pair will be, (−5,−7).

Answer 2

Answer:

$\huge \red{1 : 5 : 7}$

Step-by-step explanation:

Let roots of the equations be a , b , c and a, b , d respectively . Note that

$c \: \neq \: d$

Now relation between roots and coefficient of cubic

$ab + bc + ca = p \\ ab + bd + ad = p$

Subtract both

$(a + b)(c - d) = 0 \\ since \: \: c \neq d \: \: so \\ \\ \implies \: a + b = 0 \: \: or \: \: b = - a$

Now sum of roots

$a + b + c = - 5 \\ \implies a - a + c = - 5 \\ \implies c = - 5$

Similarly from second equation

$d = - 7$

Now product of root

$abc = - q \implies a ( - a) c = - q \\ \ \ \implies {a}^{2} c = q\\ \\ abd = - r \implies {a}^{2} d = r$

Divide both equation

$\frac{q}{r} = \frac{c}{d} = \frac{ - 5}{ - 7} = \frac{5}{7} \\ \\ \implies r = \frac{7q}{5}$

For p again take result

$ab + bc + ac = p \\ \\ \implies - {a}^{2} - ac + ac = p \\ \\ \implies p = - {a}^{2} \\ using \: \: {a}^{2} c = q \\ \\ \implies p = \frac{ - q}{c} = \frac{ - q}{ - 5} = \frac{q}{5}$

Now ratio

$p : q : r = \frac{q}{5} :q : \frac{7q}{5} \\ \\ \implies p : q : r = 1 : 5 : 7$