Important Points in Quadratic Equation and Expression

 

Important Points in Quadratic Equation and Expression

It contains mainly three Parts

I.                    Quadratic Equation

II.                  Quadratic Expression

III.                Polynomial Equation

 

1.       Polynomial

(i)                  Definition, Leading Coefficient, Degree

(ii)                Coefficient Based Polynomial Types

(iii)               Degree Based Polynomial Types

(iv)              Polynomial Function of Root if one value of Variable given then Finding value of the polynomial

(v)                Roots Transformation Method

(vi)              Theory of Equations

2.       Difference between Roots and Solutions

3.       Equation

(i)                  Equivalent Equation

(ii)                Identical Equation

4.       Identity

(i)                  Definitions, Examples

5.       Quadratic Equation

(i)                  Purely Quadratic Equation

(ii)                Complete Quadratic Equation

(iii)               Relation Between Roots and Coefficients

(iv)              Finding Roots

(v)                Sum of Roots

(vi)              Product of Roots

(vii)             Difference of Roots

(viii)           Roots given then finding Quadratic Equation

(ix)              Every Root Satisfying its Equation

(x)                Roots Transformation Method

(xi)              Newton’s Theorem

(xii)             Nature of Roots(Coefficients are real)

(a)    Condition for Real Roots

(i)                  Condition for Equal Roots

(ii)                Condition for different roots

1.       Condition for Irrational Roots

2.       Condition for Rational Roots

(I)                 Condition for Integral Roots  

(b)   Condition for Imaginary Roots

(xiii)           Roots under particular Cases

(a)    Roots are opposite in sign

(b)   Roots are equal in magnitude but opposite in sign

(c)    One root is zero

(d)   Roots are reciprocal to each other

(e)   Both roots Positive

(f)     Both roots negative

(g)    At least one root is positive

(h)   At least one root is negative

(i)      One root is 1

(j)     Greater root in magnitude is negative

(k)    Greater root in magnitude is positive

(l)      Exactly one root is infinity

(m) Both roots are at infinity

6.       Symmetric Function of Roots

7.       Cubic Equation

(i)                  Sum of Roots

(ii)                Sum of Products of Two Roots

(iii)               Product of all Roots

(iv)              Relation between two of roots given then finding arbitrary constants

(v)                If roots are in AP/GP/HP then finding roots of known equation

(vi)              Condition for all real and different roots

(vii)             Condition for three real roots with one repeated twice

(viii)           Condition for only one real root

8.       Two Quadratic Equations

(i)                  Relation between its roots

(a)    Condition for At least Two Real Roots

(b)   Condition for At least Two imaginary Roots

(c)    Condition for exactly two Real Roots or Exactly two imaginary roots

(d)   Condition for all four Real Roots

(e)   Condition for all four imaginary Roots

(ii)                Condition for Common Roots

(a)    Condition for both Common Roots

(b)   Condition for may have a common Roots

(c)    Condition for Exactly one common roots

(d)   If one common roots given between two equation then finding quadratic equation with other roots

(e)   Condition for pairwise common roots between three quadratic equations

(f)     Common roots in Polynomial equations

9.       General Quadratic Expression in two variables

(i)                  Condition so that it may be resolved into two linear factors

(ii)                Finding two Linear Factors

(iii)               Finding Range of the variables

10.   Quadratic Expression

(i)                  Its Graph as Vertical parabola with vertex

(ii)                Condition for upward and downward parabola

(iii)               Condition for quadratic expression having same sign for all real values of the variable

(iv)              Finding sign of c with parabola cutting y axis w.r.t. origin

(v)                Finding sign of b w.r.t. vertex lying on which side of origin

(vi)              Finding sign of discriminant as parabola position with x axis

(vii)             Drawing Graph of a quadratic expression

11.   Range of quadratic expression

(i)                  Finding Range for all real value of x

(a)    Finding minimum value of x if coefficient of x² is positive

(b)   Finding maximum value of x if coefficient of x² is negative

(ii)                Finding Range if x belongs to given closed interval

(iii)               Finding Range if x does not belongs to given closed interval

12.   Range of Rational Expression

(i)                  Finding Range of Rational Expression

(ii)                Finding arbitrary constant if rational expression can take all real values for all real values of the variable

13.   Location of Roots

(i)                  Condition for Both Roots more than finite number k

(ii)                Condition for Both Roots less than finite number k

(iii)               Condition for Finite number k lies between roots

(iv)              Condition for Both roots lying in an interval

(v)                Condition for Exactly one root lying in an interval

(vi)              Condition for one root lying in one interval and other root lying in other interval

14.   Common roots in Polynomial Equation

15.   Remainder theorem

16.   Intermediate value theorem

17.   Descarte’s Rule of sign

18.   Rolle’s Theorem

19.   Some important Polynomial Equations Reducible to Quadratic Equation