Discriminant of Quadratic Equations: Difference or Special Case?

[Sumedh]

Is the discriminant, of the quadratic equations, the difference between the two roots?
Or is it a special case?

 

[Hootenanny]

Is the discriminant, of the quadratic equations, the difference between the two roots?
Or is it a special case?

I'm not sure what your asking here, but the quadratic discriminant is $\Delta = b^2 - 4ac$. The two roots are

$$x_\pm = \frac{-b\pm\sqrt{\Delta}}{2a},$$

with the difference being

$$x_+ - x_- = \frac{-b + \sqrt{\Delta} + b +\sqrt{\Delta}}{2a} = \frac{\sqrt{\Delta}}{a}.$$

So in general, the discriminant is not the difference between the two roots. The condition for the discriminant to be the difference between the two roots is

$$\Delta = \frac{\sqrt{\Delta}}{a}\text{ or } \Delta = 0\;, \Delta = a^{-2}.$$

The first corresponds to the case when you have repeated roots (obviously) and the second occurs when $a^2b^2 - 4a^3c - 1 = 0$.

 

[sankalpmittal]

I'm not sure what your asking here, but the quadratic discriminant is $\Delta = b^2 - 4ac$. The two roots are

$$x_\pm = \frac{-b\pm\sqrt{\Delta}}{2a},$$

with the difference being

$$x_+ - x_- = \frac{-b + \sqrt{\Delta} + b +\sqrt{\Delta}}{2a} = \frac{\sqrt{\Delta}}{a}.$$

So in general, the discriminant is not the difference between the two roots. The condition for the discriminant to be the difference between the two roots is

$$\Delta = \frac{\sqrt{\Delta}}{a}\text{ or } \Delta = 0\;, \Delta = a^{-2}.$$

The first corresponds to the case when you have repeated roots (obviously) and the second occurs when $a^2b^2 - 4a^3c - 1 = 0$.

Yes Hooteny .

Let x and y be the two distinct roots of quadratic equation ax²+bx+c = 0
and D = b²-4ac then xy (Product of two roots)= c/a and x+y (Sum of two roots) = -b/a .

So we can also write a quadratic equation in this form :

x²+bx/a+c/a = 0
or

A quadratic equation is written in this form :
x² - (Sum of two roots)x + (Product of two roots) = 0

The only relation which establishes between equal roots of two different quadratic equations are :

c₁/a₁ = c₂/a₂ = ... = c_n/a_n

and

-b₁/a₁ = -b₂/a₂ = ... = -b_n/a_n



As Hooteny marks :

Difference of two roots of a quadratic equation is : sqrt(D)/a which is not equal to D. Discriminant (D or Δ) or determinant just determines the nature of roots of a quadratic equation.

 

[Sumedh]

Thank you very much.