Question

Сатре о

Answer 1

Answer:

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

Step-by-step explanation:

Given

$ax^2 +bx + c = 0$

Required

Condition that the roots is in m : n

Let the roots of the equation be represented as: mA and nA

A quadratic equation has the form:

$x^2 + (sum\ of\ roots)x + (product\ of\ roots)=0$

or

$x^2 - (\frac{b}{a})x + \frac{c}{a} = 0$

We have the roots to be mA and nA.

So, the sum is represented as:

$Sum = mA + nA$

$Sum = A(m + n)$

And the product is represented as:

$Product = mA * nA$

$Product = mnA^2$

By comparing:

$x^2 + (sum\ of\ roots)x + (product\ of\ roots)=0$

with

$x^2 - (\frac{b}{a})x + \frac{c}{a} = 0$

$Sum = -\frac{b}{a}$

$Product = \frac{c}{a}$

So, we have:

$Sum = -\frac{b}{a}$

$A(m + n) = -\frac{b}{a}$

Make A the subject:

$A = \frac{-b}{a(m+n)}$

$Product = \frac{c}{a}$

$mnA^2 = \frac{c}{a}$

Substitute $A = \frac{-b}{a(m+n)}$

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

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

Multiply both sides by a

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

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

Cross Multiply:

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

Hence, the condition that the ratio is in m:n is

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