Answer:
[tex]mnb^2 = ac(m+n)^2[/tex]
Step-by-step explanation:
Given
[tex]ax^2 +bx + c = 0[/tex]
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:
[tex]x^2 + (sum\ of\ roots)x + (product\ of\ roots)=0[/tex]
or
[tex]x^2 - (\frac{b}{a})x + \frac{c}{a} = 0[/tex]
We have the roots to be mA and nA.
So, the sum is represented as:
[tex]Sum = mA + nA[/tex]
[tex]Sum = A(m + n)[/tex]
And the product is represented as:
[tex]Product = mA * nA[/tex]
[tex]Product = mnA^2[/tex]
By comparing:
[tex]x^2 + (sum\ of\ roots)x + (product\ of\ roots)=0[/tex]
with
[tex]x^2 - (\frac{b}{a})x + \frac{c}{a} = 0[/tex]
[tex]Sum = -\frac{b}{a}[/tex]
[tex]Product = \frac{c}{a}[/tex]
So, we have:
[tex]Sum = -\frac{b}{a}[/tex]
[tex]A(m + n) = -\frac{b}{a}[/tex]
Make A the subject:
[tex]A = \frac{-b}{a(m+n)}[/tex]
[tex]Product = \frac{c}{a}[/tex]
[tex]mnA^2 = \frac{c}{a}[/tex]
Substitute [tex]A = \frac{-b}{a(m+n)}[/tex]
[tex]mn(\frac{-b}{a(m+n)})^2 = \frac{c}{a}[/tex]
[tex]mn\frac{b^2}{a^2(m+n)^2} = \frac{c}{a}[/tex]
Multiply both sides by a
[tex]a * mn\frac{b^2}{a^2(m+n)^2} = \frac{c}{a} * a[/tex]
[tex]\frac{mnb^2}{a(m+n)^2} = c[/tex]
Cross Multiply:
[tex]mnb^2 = ac(m+n)^2[/tex]
Hence, the condition that the ratio is in m:n is
[tex]mnb^2 = ac(m+n)^2[/tex]
