Prove that, for all values of k, the roots of the quadratic polynomial x2−(2+k)x−3 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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