Monday, 3 July 2017

algebra precalculus - Showing that roots of a quadratic polynomial are of opposite signs.




Prove that, for all values of k, the roots of the quadratic polynomial x2(2+k)x3 are real. Show further that the roots are of opposite signs.




For the first part I was able to demonstrate such by using the discriminant of the quadratic, then using the discriminant of the discriminant.



For the second part I was not able to demonstrate such.



Answer



A general quadratic with roots α and β can be written
x2(α+β)x+αβ=0


The last, constant term is the product of the roots. In your case that equals 3. The roots must therefore have opposite signs.


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