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Eddi Din [679]
3 years ago
7

Сатре о

Mathematics
1 answer:
allsm [11]3 years ago
7 0

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

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