JEE 2020 Mathematics : Quadratic Equation
JEE Main Short Notes
Quadratic Equations is one of the basic topics for your JEE Main/JEE Advanced preparation. The concept of roots of any polynomial is a basic pre-requisite for further topics in Mathematics like Trigonometry, Calculus and Coordinate Geometry. Every year there are 2-3 questions directly asked on the concepts of quadratic equations and polynomials.
1. Quadratic Equation and Roots
A polynomial equation of second degree i.e., an equation of the form ax2 + bx + c =0 where a,b,c are real numbers and a≠0, is known as a quadratic equation in x.
A quantity α is known as the root of the quadratic equation ax2 + bx + c = 0 if aα 2 + bα + c = 0
A quadratic equation cannot have more than 2 roots.
The quantity b2 -4ac is called the discriminant of the quadratic equation and denoted by D. The nature of D will determine the nature of the roots of the equation.
Case 1: D>0 : The equation ax2 + bx + c = 0 will have two distinct real roots which are .
Case 2: D=0 : The equation ax2 + bx + c = 0 has two equal real roots which are - b/2a and - b/2a.
Case 3: D<0: The equation ax2 + bx + c = 0 will have no real roots. It will have imaginary roots.
Case 4: D is a square of a rational number: The equation ax2 + bx + c = 0 will have rational roots.
Case 5: D is not a square of a rational number:The equation ax2 + bx + c = 0 will have non - rational roots i.e., irrational roots and they exist in conjugate pairs.
Few important points to keep in mind
a) If p + iq is a root, then another root will be p - iq (i=√-1 )
b) Imaginary roots always occur in pairs for any polynomial with real coefficients
c) Relation between roots and coefficients for Quadratic Equation: In a quadratic equation ax2 + bx + c = 0
d) The sum of the roots: α + β = (-b/a) = - (coefficient of x / coefficient of x2)
The product of the roots αβ =(c/a)= (constant term / coefficient of x2)
e) The quadratic equation can be wrote using the sum of roots and product of the roots as follows:
x2 – (sum of the roots) x + (product of the roots) = 0
If the product of roots of a quadratic equation is negative, then the roots are of opposite sign
If in a quadratic equation, ax2 + bx + c = 0 ; a=1 and b, c are integers and roots are rational, then the roots are integers.
2. Symmetric Function of Roots
An expression in α, β is called a symmetric function of α and β if the function is not affected by interchanging α and β. A symmetric function of α and β can always be expressed as a function of α+β and αβ.
a) α2 + β2 = (α+β)2 - 2αβ
b) α - β=±√[(α+β)2 - 4αβ]
3. Common Roots
Let a1x2 + b1x + c =0 and a2x2 + b2x + c =0 be two quadratic equations.
Case 1: When one root α is common
Case 2: When both the roots are common
4. Nature of roots of simultaneous quadratic equations
Let D1 and D2 be the discriminant of two quadratic equations. If
D1+D2 ≥ 0, then at least one of the equations must have real roots
D1+D2 < 0, then at least one of the equations must have non-real roots
D1D2 > 0, then either both the equations have real and distinct roots, or both the equations have non-real roots.
D1D2 < 0, then one of the equations has real and distinct roots while the other has non-real roots.
D1D2 = 0, then one equation has equal roots. The other equation can have both real or non-real roots.
5. Sign of roots
Let the roots of ax2 + bx + c=0 be α and β
If both roots are positive, then a and c must have the same sign
If both roots are negative, a, b, c have the same sign
If one root is positive while the other is negative then, a and c must have different signs
If roots are equal in magnitude but opposite in sign, then b=0
If the roots are reciprocal to each other than a is equal to c
If c=0, then one of the roots must be 0
If x is replaced by 1/x, then the new roots of the equation will be 1/α and 1/β
If x is replaced by x2 , then the new roots of the quadratic equation will be α, -α, β , -β
6. The relation between roots and coefficients
Let a0xn + a1xn-1 + a2xn-2 +....+an-1x + an = 0, with a≠0 be a polynomial equation. Let the roots of the equation be α1, α2 , ...,αn .
Sum of the roots taken one at a time:α1+ α2 + ...+ αn = - a1 / a0
Sum of the roots taken two at a time:α1α2 + α2α3 +........+α1αn +α2α3 +...+αn-1αn = a2 / a0
Sum of the roots taken three at a time:α1α2α3 + α1α2α4 + ...+αn-2αn-1αn = - a3 / a0
and so on
Sum of the roots taken n at a time= (-1)n-1an-1 / a0
Product of the roots , α1α2α3 ..αn-2αn-1αn = (-1)nan / a0
7. Remainder Theorem, Factor Theorem, Divisibility Theorem
The remainder theorem states that if a polynomial is f(x) is divided by (x-a) where a is independent of x, then the remainder will be f(a).
Factor theorem states that if f(a)=0, then (x-a) will be a factor of f(x)
Divisibility theorem states that if f(a)=0, then f(x) will be divisible by (x-a)
Let f(x) be divided by (x-a)
f(x)=(x-a)p(x) +R where R is the remainder
Put x=a, R=f(a)
This proves the remainder theorem.
If f(a)=0, then R=0 so (x-a) is a factor of f(x). This proves the factor theorem and divisibility theorem.
8. Inequalities Using Wavy Curve Method
Here n1, n2 ,....,nk ,m1, m2,.....mp are all natural numbers and a1,a2,.......ak, b1,b2,...bp are all real numbers and none of them are equal to each other.
a) Function Zero: A point x=a is called a function zero if f(a)=0
b) Point of discontinuity: A point x=b, is called a point of discontinuity if f(b) does not exist, that is the denominator becomes 0
c) Single Point: Consider (x -ak)nk . If nk is an odd integer then x=ak is called a single point. The function changes the sign on either side of ak
d) Double Point: Consider (x -ap)np . If np is even integer then x=ap is called a double point. The function has the same sign on either side of ap
Here a1 ,a2, a3, a4, a5, a6, a7 are function zeroes
a1 ,a2, a4, a5, a6 are single points because the function changes the sign.
a3 and a7 are double points.
9. Quadratic Expressions
Case 1: a>0
Case 2: a<0
10. Sign of quadratic expression
Let f(x)=ax2 + bx + c be a quadratic expression, with a not equal to 0. Let D be the discriminant of the corresponding quadratic equation and α and β be its roots.
(a) If D<0, the sign of f(x) is same as that of a for all values of x – either positive or negative
(b) If D=0, the sign of f(x) is same as that of a for all values of x
(c) If D>0, the sign of f(x) is same as that of a for x<α and x<β . The sign of f(x) is opposite to that of a for α<x<β
11. Location of roots
a) The root lies in (0, p) if and only if -b/a > 0 and c/a >0 when p>0
b) The root lies in (-p,0) if and only if -b/a < 0 and when p>0
c) Both roots are greater than a given number k if the following three conditions are satisfied, D≥0 , (-b/2a)>k and af(k)>0
d) Both roots are less than a given number k if the following three conditions are satisfied,D≥0 , (-b/2a)<k and af(k)>0
e) Both the roots will lie in the interval (k1,k2) if D≥0, k1 < -b/2a < k2 and af(k2)>0
f) Exactly one root will lie in (k1,k2) if f(k1)·f(k2)<0
g) A given number k will lie in between the roots if af(k)<0
12. Highest and Least Values of a quadratic expression
If a<0, then the highest value of f(x)=ax2 + bx + c is -D/4a and it is obtained at x=-b/2a
If a>0, then the least value of f(x) is -D/4a and it is obtained at x=-b/2a
All the best,