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.

 

[CellCoree]

The discriminant of a quadratic equation is a special case that provides information about the nature of the solutions to the equation. It is not simply the difference between the two roots. The discriminant is a value that is calculated from the coefficients of the quadratic equation and can be used to determine whether the equation has two distinct real roots, two complex roots, or one repeated root. It can also indicate whether the roots are rational or irrational numbers. Therefore, the discriminant is an important tool in understanding the behavior of quadratic equations and is not simply a difference between the two roots.