Alpha and Beta

Quadratics

Alpha and Beta

Roots And Coefficients

 $ax^2+bx+c=0$  
The coefficients are a, b and c.

 $\propto\ \beta$  - these are called roots
 $\propto-\ alpha\ \ \ \beta-beta$  
Note: the symbol for alpha can also be represented by  $\alpha$  

There are relationships between coefficients of a quadratic equation and the roots alpha and beta. The relationship is based on the Sum of the Roots and the Product of the roots.

Representation of the relationships.

Find the values of x if  $x^2+4x-12=0$  

 $x^2+4x-12=0$  (factorize)
 $\left(x+6\right)\left(x-2\right)=0$  
 $\therefore x=-6\ or\ x=2$  

Let roots be  $\propto\ \&\ \beta$  
 $\propto=-6\ \ \beta=2$           Sum of Roots :   $\propto+\beta=-6+2=-4$                                                   Product of Roots :  $\propto\beta=-6\left(2\right)=-12$  
Therefore the relationship between the roots and coefficients is   $\propto+\beta=\frac{-b}{a}\ \ \&\ \ \propto\beta=\frac{c}{a}$  

example 1.

If  $x^2-2x-8=0\ has\ roots\propto and\ \beta$  find: a. $\propto+\beta\ \ b.\propto\beta$  

 $x^2-3x-8=0$  
 $a=1\ \ b=-2\ \ c=-8$  

a)  $\propto+\beta=\frac{-b}{a}=\frac{-\left(-2\right)}{1}=\frac{2}{1}=2$  
b)  $\propto\beta=\frac{c}{a}=\frac{-8}{1}=-8$  

Example 2.

If  $\propto and\$  are the roots of the equation  $2x^2-5x+3=0$  find:

 1.  $\propto+\beta$   2.  $\propto\beta$  

 $2x^2-5x+3=0$  
 $a=2\ \ b=-5\ \ c=3$  

 $\propto+\beta=\frac{-b}{a}=\frac{-\left(-5\right)}{2}=\frac{5}{2}$  
 $\propto\beta=\frac{c}{a}=\frac{3}{2}$  

Operations with alpha and beta.

1.  $\propto+\beta=\frac{-b}{a}$    
2.  $\propto\beta=\frac{c}{a}$  
3.  $\propto^2\beta^2=\left(\propto\beta\right)^2$  
4.  $\propto^2\beta+\beta^2\propto=\propto\beta\left(\propto+\beta\right)$  
5. $\frac{x}{\propto}+\frac{x}{\beta}=\frac{x\propto+x\beta}{\propto\beta}=\frac{x\left(\propto+\beta\right)}{\propto\beta}$  where x is a constant (e.g 2,3,4,5)

6.  $\left(\propto+\beta\right)^2=squared\ of\ \propto+\beta$  
7.  $\frac{\propto}{\beta}+\frac{\beta}{\propto}=\frac{\propto^2+\beta^2}{\propto\beta}=\ \frac{\left(\propto+\beta\right)^2-2\propto\beta}{\propto\beta}$  

Examples of the different operations.

If  $\propto and\ \beta$  are the roots of the equation  $x^2-3x-18=0$  find:

 a.  $\propto+\beta\ \ \ \ b.\propto\beta\ \ \ \ c.\propto^2\beta^2\ \ \ d.\propto^2\beta+\beta^2\propto\$ 

 a) $\propto+\beta=\frac{-b}{a}=\frac{-\left(-3\right)}{1}=3$  

b) $\propto\beta=\frac{c}{a}=\frac{-18}{1}=-18$  

c)  $\propto^2\beta^2\Longrightarrow\left(\propto\beta\right)^2=\left(-18\right)^2=324$  
d)  $\propto^2\beta+\beta^2\propto\Longrightarrow\propto\beta\left(\propto+\beta\right)=-18\left(3\right)=-54$  

more examples/continued

If  $\propto and\ \beta$  are the roots of the equation  $3x^2-x-5=0$  . Find the following a. $\propto+\beta$   b. $\propto\beta$    c. $\propto^2+\beta^2\$   d. $\frac{6}{\propto}+\frac{6}{\beta}$   e. $\left(\propto\beta\right)^2$  

a. $\propto+\beta=\frac{-b}{a}=\frac{-\left(-1\right)}{3}=\frac{1}{3}$  
b.  $\propto\beta=\frac{c}{a}=\frac{-5}{3}$  
c.  $\propto^2+\beta^2=\left(\propto+\beta\right)^2-2\propto\beta=\left(\frac{1}{3}\right)^2-2\left(-\frac{5}{3}\right)$          
=   $\frac{1}{9}+\frac{10}{3}=\frac{31}{9}$  

d.  $\frac{6}{\propto}+\frac{6}{\beta}=\frac{6\beta+6\propto}{\propto\beta}=\frac{6\left(\beta+\propto\right)}{\propto\beta}$  

 $=\frac{6\left(\frac{1}{3}\right)}{-\frac{5}{3}}=\frac{2}{-\frac{5}{3}}=-\frac{6}{5}$  

Note :
 $\frac{2}{1}\div-\frac{5}{3}=\frac{2}{1}\times-\frac{3}{5}=-\frac{6}{5}$     and  $\beta+\propto\equiv\propto+\beta$  

e. $\left(\propto+\beta\right)^2=\left(\frac{1}{3}\right)^2=\frac{1}{9}$  

New Roots

Steps:

1. State coefficients a,b,c .
2. Find  $\propto+\beta,\ \propto\beta$  
3. For new roots find: sum of roots and product of roots.
4. Substitute  $\propto+\beta,\ \propto\beta\$  from step 2 into  $x^2-\left(sum\right)x+\ product\ =0$  
N.B your answer will be a quadratic equation.

Common new roots

 $\cdot\propto+1\ ,\ \beta+1$  
 $\cdot\frac{x}{\propto},\ \frac{x}{\beta}$  where x is a constant 
 $\cdot\propto-1\ ,\ \beta-1$  
 $\cdot\frac{1}{\propto},\ \frac{1}{\beta}$  
 $\cdot\frac{\propto}{\beta},\ \frac{\beta}{\propto}$  

Eg 1

If  $\propto\ and\ \beta$  are the roots of  $x^2+4x-12=0$  find an equation who's roots are  $\propto+1\ and\ \ \beta+1$  .

 $x^2+4x-12=0\ \ \ \ \ a=1,\ b=4,\ c=-12$   $\propto+\beta=-\frac{b}{a}=\frac{-4}{1}=-4$  
 $\propto\beta=\frac{c}{a}=\frac{-12}{1}=-12$  
 $Sum:\ \propto+1\ +\beta+1=\propto+\beta+2=-4+2=-2$   $\Pr oduct\ :\ \left(\propto+1\right)\left(\beta+1\right)=\propto\beta+\propto+\beta+1$                                                                                        $=-12-4+1=-15$  
Equation  $\Longrightarrow x^2-\left(-2\right)x+\left(-15\right)=0$                                                                     $=x^2+2x-15=0$  

Eg 2

If  $\propto\ and\ \beta$  are the roots of the equation  $x^{2\ }-2x-8=0$  find the equation with roots  $\frac{3}{\propto}\ and\ \frac{3}{\beta}$  .

 $x^2-2x-8=0\$   a=1, b=-2, c=-8
 $\propto+\beta=\frac{-b}{a}=\frac{-\left(-2\right)}{1}=2$  
 $\propto\beta=\frac{c}{a\ }=\frac{-8}{1}=-8$  
 $Sum:\ \frac{3}{\propto}+\frac{3}{\beta}=\frac{3\beta+3\propto}{\propto\beta}=\frac{3\left(\propto+\beta\right)}{\propto\beta}=\frac{3\left(2\right)}{-8}=-\frac{6}{8}$   $\Pr oduct:\ \frac{3}{\propto}\times\frac{3}{\beta}=\frac{9}{\propto\beta}=\frac{9}{-8}=-\frac{9}{8}$  
 $Equation\ \Longrightarrow\ x^2-\left(-\frac{6}{8}\right)x+\left(-\frac{9}{8}\right)=0\ \left(\times8\right)$   $=8x^2+6x-9=0$  

Eg 3

If  $\propto and\ \beta$  are the roots of the equation  $2x^2+5x-3=0$  .  Find the equation who's roots are  $\propto-1\ and\ \beta-1$  .

 $2x^2+5x-3=0$     $\propto+\beta=\frac{-b}{a}=-\frac{5}{2}$  
a=2, b=5 , c=-3               $\propto\beta=\frac{c}{a}=-\frac{3}{2}$  
 $Sum:\ \propto-1+\beta-1=\propto+\beta-2=-\frac{5}{2}-2=-\frac{9}{2}$   $\Pr oduct:\ \left(\propto-1\right)\left(\beta-1\right)=\propto\beta-\propto-\beta+1$   $=\propto\beta-1\left(\propto+\beta\right)+1=-\frac{3}{2}-1\left(-\frac{5}{2}\right)+1=2$ 
Equation
 $\Longrightarrow x^2-\left(-\frac{9}{2}\right)x+\left(2\right)=0\left(\times2\right)=x^2+9x+4=0$ 

Eg4

If  $\propto and\ \beta$  are the roots of the equation  $x^2+2x-1=0$  find a quadratic equation who's roots are  $\frac{1}{\propto}and\ \frac{1}{\beta}$  .
 $x^2+2x-1=0$                 $\propto+\beta=\frac{-b}{a}=\frac{-2}{1}=-2$  
a=1 , b=2 , c=-1                       $\propto\beta=\frac{c}{a}=\frac{-1}{1}=-1$  
 $Sum:\ \frac{1}{\propto}+\frac{1}{\beta}=\frac{\beta+\propto}{\propto\beta}=\frac{\left(\propto+\beta\right)}{\propto\beta}=\frac{-2}{1}=-2$   $\Pr oduct:\frac{1}{\propto}\times\frac{1}{\beta}=\frac{1}{-1}=-1$  
Equation  $\Longrightarrow x^2-\left(-2\right)x+\left(-1\right)=0$  
 $x^2+2x-1=0$  

Eg 5

If  $\propto\ and\ \beta$  are the roots of the equation  $2x^2+3x-1=0$  .Find the equation who's roots are  $\frac{\propto}{\beta}and\frac{\beta}{\propto}$  .
 $2x^2+3x-1=0$         $\propto+\beta=\frac{-b}{a}=-\frac{3}{2}$  
a=2, b=3 c=-1                         $\propto\beta=\frac{c}{a}=\frac{-1}{2}=-\frac{1}{2}$  
 $Sum:\frac{\propto}{\beta}+\frac{\beta}{\propto}=\frac{\propto^2+\beta^2}{\propto\beta}=\frac{\left(\propto+\beta\right)^2-2\propto\beta}{\propto\beta}$   $=\frac{\left(-\frac{3}{2}\right)^2-2\left(-\frac{1}{2}\right)}{-\frac{1}{2}}=-\frac{13}{2}$     E: $x^2-\left(-\frac{13}{2}\right)x+\left(1\right)=0\ \left(\times2\right)$  
 $\Pr oduct:\left(\frac{\propto}{\beta}\right)\left(\frac{\beta}{\propto}\right)=1\$                  $2x^2+13x+2=0$