maths help

Question

condition for a quadratic equation to have one positive and 1 negative root is

Answer 1

     I'm sure you know the quadratic formula so the numerator determines the sign of the entire fraction since it has the + or - . So we want to insure that -b + sqrt b^2-4ac and -b - sqrt b^2-4ac have opposite signs. Let's take some real examples so we can get a handle on the problem before generalizing.

4 + 7 and 4-7 comes out to 11 and -3    opposite signs so good

4 +1 and 4-1 comes out  to 5 and 3        same sign so bad

-5 + 3 and -5 - 3 comes out to -2 and -8   same sign so bad

-5 +9 and -5 -9 comes out to 4 and -14    opposit signs so good

 

What's so special about case 1 and case 4? The second number is bigger than the first? Be careful before you make generaliziations!!! Case 3 also has this property...3 is bigger than -5 but 3 is not bigger than | 5 | ...... the absolute value of 5. So the second number must always be bigger than the absolute value of the first. So,

sqrt b^2-4ac must be greater than the absolute value of -b. That is b^2-4ac > |-b|.

 

The only other condition that must hold is that b^2-4ac must be positive otherwise you get equal roots or complex roots.