A variant of the quadratic formula is

**x** = (2

**c**)/(-

**b** +/- sqrt[

**b**^2-4

**ac**])

From this it is easy to see that for a variable **a** that one root is always greater than -**c**/**b** and the other is always less than -**c**/**b**.

Also note that from this form of the quadratic equation, lim {**a** -> 0} (2**c**)/(-**b** +/- sqrt[**b**^2-4**ac**]) is equal to -**c**/**b** or diverges to infinity, depending on which root is examined.

Let **p1** and **p2** be the roots of **a**x^2+ **b**x + **c** = 0.

Let **q1** and **q2** be the roots of -**a**x^2+ **b**x + **c** = 0.

Let **r1** and **r2** be the roots of (**a**/2)x^2+ **b**x + **c** = 0.

Without loss of generality, **p2** < -**c**/**b** < **p1**; **q2** < -**c**/**b** < **q1**; and **r2** < -**c**/**b** < **r1**.

From the limit earlier, exactly one of **r1** and **r2** approaches -**c**/**b**. If that root is **r1** then **q2** < -**c**/**b** < **r1** < **p1**. Also, if that root is **r2** then **p2** < **r2** < -**c/b** < **q1**.

Therefore (**a**/2)x^2+ **b**x + **c** = 0 has root between some **p** and **q**. More generally this is true for any **k** with abs(**a**)>abs(**k**)>0.

*Edited on ***July 14, 2016, 1:56 pm**