Question

Prove that if ax³+bx²+cx+d=0 has three real and distinct roots, then b²> 3ac.​

Answer 1

$\large\underline{\sf{Solution-}}$

We know that,

For a cubic equation, ax³+bx²+cx+d=0, it always have one real root.

To check the nature of remaining 2 roots, we use the concept of differentiation.

Differentiate the given cubic equation.

So, it reduced to quadratic equation.

Now, solve the quadratic to get the roots.

Then check the maxima and minima at these points.

Three cases arises :-

Case :- 1 If maxima and minima value are of opposite sign, the cubic equation has three real and distinct roots.

Case :- 2 If one value is 0, then equation has 2 equal roots.

Case :- 3 If both the values are 0, then equation has one real and two complex roots.

Here,

The given cubic equation is

$\rm :\longmapsto\: {ax}^{3} + {bx}^{2} + cx + d = 0$

has three real and distinct roots.

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\: {3ax}^{2} + {2bx}^{} + c = 0$

For roots to be real and distinct,

Discriminant > 0

$\rm :\longmapsto\: {(2b)}^{2} - 4(3a)(c) > 0$

$\rm :\longmapsto\: {4b}^{2} - 12ac > 0$

$\rm :\longmapsto\: {b}^{2} - 3ac > 0$

$\rm :\longmapsto\: {b}^{2} > 3ac$

Hence, Proved

Answer 2

Solution

We know that,

For a cubic equation, ax³+bx²+cx+d=0, it always have one real root.

To check the nature of remaining 2 roots,

we use the concept of differentiation.

Differentiate the given cubic equation.

So, it reduced to quadratic equation.

Now, solve the quadratic to get the roots.

Then check the maxima and minima at these points.

Three cases arises :

Case - 1 - If maxima and minima value are of opposite sign, the cubic equation has three real and distinct roots.

Case - 2 - If one value is O, then equation has 2 equal roots.

Case - 3 - If both the values are 0, then equation has one real and two complex roots.

Here,

The given cubic equation is

ax³ + bx² + cx + d = 0

has three real and distinct roots.

On differentiating both sides w. r. t. x, we get

: 3ax² +2bx + c = 0

For roots to be real and distinct,

Discriminant > 0

(2b)² - 4(3a) (c) > 0

= 4b² - 12ac > 0

= b²-3ac > 0

= b² > 3ac

Hence, Proved