Question

if the roots of the equation ax²+bx+c=0 are in the ratio m:n, prove that mnb²=ac(m+n)²

Answer 1

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Answer 2

Answer:

If the roots of the equation ax²+bx+c=0 are in the ratio m:n then mnb²=ac(m+n)².

Step-by-step explanation:

The given equation is

$ax^2+bx+c=0$

It is given that the roots of the equation ax²+bx+c=0 are in the ratio m:n.

Let the roots of given equation are mx and nx.

We know that the sum of roots is equal to -b/a and the product of roots is c/a.

$mx+nx=b\Rightarrow x(m+n)=-\frac{b}{a}\Rightarrow x=-\frac{b}{a(m+n)}$         ....(1)

$mx\times nx=\frac{c}{a}\Rightarrow mnx^2=\frac{c}{a}$                                             .....(2)

Substitute the value of x from equation (1) in equation (2).

$mn(-\frac{b}{a(m+n)})^2=\frac{c}{a}$

$mnb^2=\frac{a^2c(m+n)^2}{a}$

$mnb^2=ac(m+n)^2$

Hence proved.