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Ranging roots (Posted on 2023-06-30)

For what values of m does the equation x² + (2m + 6)x + 4m + 12 = 0 have two real roots, both of them greater than -1.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

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The discriminant of this quadratic is 4m^2+8m-12 = 4(m-1)(m+3).

So the equation only has two real roots if m>1 or m<-3

The solutions can be written as -m-3 +/- sqrt(m^2+2m-3)

If m>1 the smaller of these will clearly be less than -1.

If m<-3 we require the smaller solution greater than -1

-m-3-sqrt(m^2+2m-3) > -1

m > -3.5

The answer to the question (which is missing a ?) is -3.5<m<-3.

Posted by Jer on 2023-06-30 09:44:08

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