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# what is the value of the discriminant?

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Guest Feb 22, 2017
edited by Guest  Feb 22, 2017

#2
+18630
+25

what is the value of the discriminant?

Degree 2
The quadratic polynomial $${\displaystyle ax^{2}+bx+c\,} \,$$ has discriminant $${\displaystyle b^{2}-4ac\,.}$$

The square root of the discriminant appears in the quadratic formula for the roots of the quadratic polynomial:

$${\displaystyle x_{1,2}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$$

The discriminant is zero if and only if the two roots are equal.

If a, b, c are real numbers, the polynomial has two distinct real roots if the discriminant is positive,

and two complex conjugate roots if it is negative.

Degree 3
The zero set of discriminant of the cubic  $${\displaystyle ax^{3}+bx^{2}+cx+d\,}$$  has discriminant

$${\displaystyle b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.}$$
The discriminant is zero if and only if at least two roots are equal.

If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers,

and negative if there is one real root and two complex conjugate roots.

Degree 4
The discriminant of the quartic polynomial  $${\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e\,}$$has discriminant
{\displaystyle {\begin{aligned}{}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\ &{}-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\ &{}+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\ &{}-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\,.\end{aligned}}}
The discriminant is zero if and only if at least two roots are equal.
If the coefficients are real numbers and the discriminant is non-zero,
the discriminant is negative if there is two real root and two complex conjugate roots,
and it is positive if the roots are either all real or all non-real.

heureka  Feb 22, 2017
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#1
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b^2 - 4ac     if it is lees thatn zero there are two conjugate irrational roots

if it equals zero thee are two real identical root

if it is > zero there are two different roots

ElectricPavlov  Feb 22, 2017
#2
+18630
+25

what is the value of the discriminant?

Degree 2
The quadratic polynomial $${\displaystyle ax^{2}+bx+c\,} \,$$ has discriminant $${\displaystyle b^{2}-4ac\,.}$$

The square root of the discriminant appears in the quadratic formula for the roots of the quadratic polynomial:

$${\displaystyle x_{1,2}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$$

The discriminant is zero if and only if the two roots are equal.

If a, b, c are real numbers, the polynomial has two distinct real roots if the discriminant is positive,

and two complex conjugate roots if it is negative.

Degree 3
The zero set of discriminant of the cubic  $${\displaystyle ax^{3}+bx^{2}+cx+d\,}$$  has discriminant

$${\displaystyle b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.}$$
The discriminant is zero if and only if at least two roots are equal.

If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers,

and negative if there is one real root and two complex conjugate roots.

Degree 4
The discriminant of the quartic polynomial  $${\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e\,}$$has discriminant
{\displaystyle {\begin{aligned}{}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\ &{}-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\ &{}+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\ &{}-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\,.\end{aligned}}}
The discriminant is zero if and only if at least two roots are equal.
If the coefficients are real numbers and the discriminant is non-zero,
the discriminant is negative if there is two real root and two complex conjugate roots,
and it is positive if the roots are either all real or all non-real.

heureka  Feb 22, 2017

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