# Quadratic Equations with Real Coefficients

Consider the quadratic equation

*ax*

^{2}+

*bx*+

*c*= 0 ...(1)

where

*a*,*b*,*c*âˆˆ*R*and*a*â‰ 0.Roots of the equation are given by

*x*=

Now, if we look at these roots, we observe that the roots depend upon the value of the quantity

*b*^{2}â€“ 4*ac*.This quantity is generally denoted by

*D*and is known as the discriminant of the quadratic equation (1).We also observe the following results:

*Notes:*- If
*a*,*b*,*c*âˆˆ*Q*and*b*^{2}â€“ 4*ac*is positive but not a perfect square, then roots are irrational and they always occur in conjugate pair like 2 + and 2 â€“ . However, if*a*,*b*,*c*are irrational numbers and*b*^{2}â€“ 4*ac*is positive but not a perfect square, then the roots may not occur in conjugate pairs. For example, the roots of the equation*x*^{2}â€“ (5 + )*x*+ 5= 0 are 5 and which do not form a conjugate pair. - If
*b*^{2}â€“ 4*ac*< 0, then roots of equations are complex. If*a*,*b,*and c are real then complex roots occur in conjugate pair like of the form*p*+*iq*and*p*â€“*iq*. If all coefficients are not real then complex roots are not conjugate. - Relations between roots and coefficients
*Î±*+*Î²*= =*Î±**Î²*= =*S*and product is*P*, then quadratic equation is given by*x*^{2}â€“*Sx*+*P*= 0. **Symmetric functions of roots***Î±*^{2}+*Î²*^{2}= (*Î±*+*Î²*)^{2}â€“ 2*Î±Î²*- (
*Î±*â€“*Î²*)^{2 }= (*Î±*+*Î²*)^{2}â€“ 4*Î±Î²* *Î±*^{3}+*Î²*^{3}= (*Î±*+*Î²*)^{3}â€“ 3*Î±Î²*(*Î±*+*Î²*)*Î±*+^{n}*Î²*= (^{n}*Î±*+*Î²*)(*Î±*^{n}^{â€“1}+*Î²*^{n}^{â€“1})*Î±Î²*(*Î±*^{n}^{â€“2}+*Î²*^{n}^{â€“2})*S*= (_{n}*Î±*+*Î²*)*S*_{n}_{â€“1}â€“*Î±Î²S*_{n}_{â€“2}*S*=_{n}*Î±*+^{n}*Î²*^{n }