Quadratic Equations - Nature of Roots

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• i) An expression of the form a₀xⁿ + a₁x^n-1+....+ a_n = 0, where n is a positive integer and a₀, a₁,...,a_n belong to some number system F, is called a polynomial in the variable x over F.

• ii) The degree of polynomial is defined as the highest index of the variable x occurring in the polynomial.

i) An identity is a statement of equality between two expressions which is true for all values of the variable involved.

ii) An equation is a statement of equality between two expressions which is not true for all values of the variable involved.

• If f(x)=0 is a polynomial equation and f(a)=0, then 'a' is a root of the polynomial equation.

If ax² + bx + c = 0, a = 0, then  where a, b, c be any complex numbers.

• A quadratic equation has exactly two roots.

• For the quadratic equations, we have

i) b² - 4ac > 0 Roots are real and distinctii) b² - 4ac = 0 Roots are real and equaliii) b² - 4ac < 0 Roots are imaginary and distinct

• If the roots of ax2 + bx + c = 0, a(¹ 0),b, c Î R, are a and b, then

• An expression in a, b is called a symmetric function of a, b if the expression is not affected by interchanging a and b.

• The quadratic equation with roots a and b is given by x²- Sx + P = 0 where S = a + b and P = ab.