# Difference between revisions of "User:Vqbc/Testing"

Line 1: | Line 1: | ||

− | <math>\ | + | The '''discriminant''' of a [[quadratic equation]] of the form <math>a{x}^2+b{x}+{c}=0</math> is the quantity <math>b^2-4ac</math>. When <math>{a},{b},{c}</math> are real, this is a notable quantity, because if the discriminant is positive, the equation has two [[real]] [[root]]s; if the discriminant is negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. |

+ | |||

+ | ==Discriminant of polynomials of degree n== | ||

+ | |||

+ | The discriminant can tell us something about the roots of a given polynomial <math>p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0</math> of degree <math>n</math> with all the coefficients being real. But for polynomials of degree 4 or higher it can be difficult to use it. | ||

+ | |||

+ | ===General formula of discriminant=== | ||

+ | |||

+ | We know that the discriminant of a polynomial is the product of the squares of the differences of the polynomial roots <math>r_i</math>, so, | ||

+ | |||

+ | <math>D(p)=a_n^{2n-2}\prod_{i<j}^{n}(r_i-r_j)^2</math> | ||

+ | |||

+ | ====When <math>n=2</math>==== | ||

+ | |||

+ | Given a polynomial <math>p(x)=ax^2+bx+c</math>, its discriminant is <math>D(p)=b^2-4ac</math>, wich can also be denoted by <math>\Delta=b^2-4ac</math>. | ||

+ | |||

+ | For <math>\Delta>0</math> we have the graph | ||

+ | |||

+ | |||

+ | [[Image:Delta_greater_than_0.png|thumb|center|300x300px|link=Gmass]] | ||

+ | |||

+ | wich has two distinct real roots. | ||

+ | |||

+ | For <math>\Delta<0</math> we have the graph | ||

+ | |||

+ | [[File:Delta_lower_than_0.png|thumb|center|300x300px]] | ||

+ | |||

+ | wich has two non-real roots. | ||

+ | |||

+ | And for the case <math>\Delta=0</math>, | ||

+ | |||

+ | [[Image:Delta_equal_to_0.png|thumb|center|300x300px]] | ||

+ | |||

+ | ====When <math>n=3</math>==== | ||

+ | |||

+ | The discriminant of a polynomial <math>p(x)=ax^3+bx^2+cx+d</math> is given by <math>D(p)=b^2c^2-4b^3d-4ac^3+18abcd-27a^2d^2</math>. | ||

+ | |||

+ | |||

+ | Also, the compressed cubic form <math>p(z)=z^3+pz+q</math> has discriminant <math>D(p)=-4p^3-27q^2</math>. We can compress a polynomial of degree 3, wich also makes possible to us to use Cardano's formula, by doing the substitution <math>x=z-\frac{a}{3}</math> on the polynomial <math>p(x)=x^3+ax^2+bx+c</math>. | ||

+ | |||

+ | *If <math>D=0</math>, then at least two of the roots are equal; | ||

+ | *If <math>D<0</math>, then all three roots are real and distinct; | ||

+ | *If <math>D>0</math>, then one of the roots is real and the other two are complex conjugate. | ||

+ | |||

+ | ====When <math>n=4</math>==== | ||

+ | |||

+ | The quartic polynomial <math>p(x)=ax^4+bx^3+cx^2+dx+e</math> has discriminant | ||

+ | |||

+ | <math>D(p)=256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e-27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de+18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2</math> | ||

+ | |||

+ | *If <math>D=0</math>, then at least two of the roots are equal; | ||

+ | *If <math>D>0</math>, then the roots are all real or all non-real; | ||

+ | *If <math>D<0</math>, then there are two real roots and two complex conjugate roots. | ||

+ | |||

+ | |||

+ | ====Some properties==== | ||

+ | |||

+ | For <math>n\geq4</math> we can say that | ||

+ | |||

+ | *The polynomial has a multiple root if, and only if, <math>D=0</math>; | ||

+ | *If <math>D>0</math>, with <math>k</math> being a positive integer such that <math>k\geq\frac{n}{4}</math>, with <math>n</math> being the degree of the polynomial, then there are <math>2k</math> pairs of complex conjugate roots and <math>n-4k</math> real roots; | ||

+ | *If <math>D<0</math>, with <math>k</math> being a positive integer such that <math>k\geq\frac{n-2}{4}</math>, then there are <math>2k+1</math> pairs of complex conjugate roots and <math>n-4k+2</math> real roots. | ||

+ | |||

+ | == Example Problems == | ||

+ | === Introductory === | ||

+ | * (AMC 12 2005) There are two values of <math>a</math> for which the equation <math>4x^2+ax+8x+9=0</math> has only one solution for <math>x</math>. What is the sum of these values of <math>a</math>? | ||

+ | |||

+ | Solution: Since we want the <math>a</math>'s where there is only one solution for <math>x</math>, the discriminant has to be <math>0</math>. <math>(a+8)^2-4(4)(9)=a^2+16a-80=0</math>. The sum of these values of <math>a</math> is <math>-16</math>. | ||

+ | |||

+ | === Intermediate === | ||

+ | * [[1977_Canadian_MO_Problems/Problem_1 | 1977 Canadian MO Problem 1]] | ||

+ | |||

+ | == Other resources == | ||

+ | * [http://en.wikipedia.org/wiki/Discriminant Wikipedia entry] |

## Revision as of 17:57, 5 December 2020

The **discriminant** of a quadratic equation of the form is the quantity . When are real, this is a notable quantity, because if the discriminant is positive, the equation has two real roots; if the discriminant is negative, the equation has two nonreal roots; and if the discriminant is 0, the equation has a real double root.

## Contents

## Discriminant of polynomials of degree n

The discriminant can tell us something about the roots of a given polynomial of degree with all the coefficients being real. But for polynomials of degree 4 or higher it can be difficult to use it.

### General formula of discriminant

We know that the discriminant of a polynomial is the product of the squares of the differences of the polynomial roots , so,

#### When

Given a polynomial , its discriminant is , wich can also be denoted by .

For we have the graph

wich has two distinct real roots.

For we have the graph

wich has two non-real roots.

And for the case ,

#### When

The discriminant of a polynomial is given by .

Also, the compressed cubic form has discriminant . We can compress a polynomial of degree 3, wich also makes possible to us to use Cardano's formula, by doing the substitution on the polynomial .

- If , then at least two of the roots are equal;
- If , then all three roots are real and distinct;
- If , then one of the roots is real and the other two are complex conjugate.

#### When

The quartic polynomial has discriminant

- If , then at least two of the roots are equal;
- If , then the roots are all real or all non-real;
- If , then there are two real roots and two complex conjugate roots.

#### Some properties

For we can say that

- The polynomial has a multiple root if, and only if, ;
- If , with being a positive integer such that , with being the degree of the polynomial, then there are pairs of complex conjugate roots and real roots;
- If , with being a positive integer such that , then there are pairs of complex conjugate roots and real roots.

## Example Problems

### Introductory

- (AMC 12 2005) There are two values of for which the equation has only one solution for . What is the sum of these values of ?

Solution: Since we want the 's where there is only one solution for , the discriminant has to be . . The sum of these values of is .