Quadratic Expressions-I

QUADRATIC EXPRESSIONS - I

1. Quadratic Expression:
An algebraic expression of the form ax²+bx+c, where  C here Cis the set of complex numbers, is called a quadratic expression in the variable x.
Examples:(i) 3x²-2x+5
                (ii) 2ix²-x+3
                (iii) 6y²+3, is also a quadratic expression in the variable y.

2. In the quadratic expression ax²+bx+c, ax² is called quadratic term, bx is called linear term and c is called constant term.  

3. Quadratic equation and Quadratic Function:
    If ax²+bx+c is a quadratic expression, then ax²+bx+c=0 is called quadratic equation and f(x)=ax²+bx+c is called quadratic function.
Example: (i) x² - 3x +5 is called quadratic expression
          (ii) x²-3x+5=0 is called quadratic equation
          (iii) f(x) =x²-3x+5 is called quadratic function

4.  Zero of a quadratic expression:
A complex number \( \alpha \), if exists, is said to be a zero of the quadratic expression ax²+bx+c if and only if \( a\alpha ^2 + b\alpha + c = 0 \)
Example:i) 2 is a zero of x²-5x+6 since 2² -5(2)+6=0
    ii) i is a zero of x²+1=0 since i² +1 = -1+1 =0
Note:  Every quadratic expression will have at most two zeroes.

5.    Root of a quadratic equation:
If  is a zero of ax²+bx+c, then  is called a root of ax²+bx+c=0
Example : (i) -1 is a root of x²-6x-7 = 0 
since (-1)²-6(-1)-7       =1+6-7=0

6.    Theorem:
If \( \alpha \) is a root of ax²+bx+c=0 then \( \alpha = \frac{{ - b \pm \sqrt {b^2 - 4ac} }} {{2a}}\,\,\,\,\,where\,\,a \ne 0,a,b,c \in \)C
Proof :
Given \( \alpha \) is a root of ax² +bx+c=0
we have \( a\alpha ^2 + b\alpha + c = 0 \)
 \( \Rightarrow a\left( {\alpha ^2 + \frac{b} {a}\alpha + \frac{c} {a}} \right) = 0 \) 
 \( \Rightarrow \alpha ^2 + \frac{b} {a}\alpha + \frac{c} {a} = 0 \)  since \( a \ne 0 \)
\( \Rightarrow \alpha ^2 + 2.\alpha .\frac{b} {{2a}} + \left( {\frac{b} {{2a}}} \right)^2 - \left( {\frac{b} {{2a}}} \right)^2 + \frac{c} {a} = 0 \)   
\( \Rightarrow \left( {\alpha + \frac{b} {{2a}}} \right)^2 = \frac{{b^2 }} {{4a^2 }} - \frac{c} {a} \)  
\( \Rightarrow \left( {\alpha + \frac{b} {{2a}}} \right)^2 = \frac{{b^2 - 4ac}} {{4a^2 }} \)
\( \Rightarrow \alpha + \frac{b} {{2a}} = \sqrt {\frac{{b^2 - 4ac}} {{4a^2 }}} \)
\( \Rightarrow \alpha + \frac{b} {{2a}} = \pm \frac{{\sqrt {b^2 - 4ac} }} {{2a}} \)
\( \Rightarrow \alpha = \frac{{ - b}} {{2a}} \pm \frac{{\sqrt {b^2 - 4ac} }} {{2a}} \)
\( \Rightarrow \alpha = \frac{{ - b \pm \sqrt {b^2 - 4ac} }} {{2a}} \)
Hence Proved.​​​​​​