# 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 ax^{2} + 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 ax^{2 }+ bx + c = 0 if aα^{ 2 }+ bα + c = 0

A quadratic equation cannot have more than 2 roots.

The quantity b^{2} -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 ax^{2 }+ bx + c = 0 will have two distinct real roots which are .

**Case 2: D=0 : **The equation ax^{2 }+ bx + c = 0 has two equal real roots which are - b/2a and - b/2a.

**Case 3: D<0: **The equation ax^{2 }+ 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 ax^{2 }+ bx + c = 0 will have rational roots.

**Case 5: D is not a square of a rational number:**The equation ax^{2 }+ 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 ax^{2 }+ bx + c = 0

**d) The sum of the roots: ****α**** + ****β**** = (-b/a)**** = - (coefficient of x / coefficient of x ^{2})**

**The product of the roots αβ**** =(c/a)****= (constant term / coefficient of x ^{2})**

**e)** The quadratic equation can be wrote using the sum of roots and product of the roots as follows:

**x ^{2} – (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, ax^{2 }+ 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) α ^{2} + β^{2} = (α+β)^{2} - 2αβ**

**b) α - β=±√[(α+β) ^{2} - 4αβ]**

## 3. Common Roots

Let a_{1}x^{2} + b_{1}x + c =0 and a_{2}x^{2} + b_{2}x + 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 D_{1} and D_{2} be the discriminant of two quadratic equations. If

D_{1}+D_{2} ≥ 0, then at least one of the equations must have real roots

D_{1}+D_{2} < 0, then at least one of the equations must have non-real roots

D_{1}D_{2} > 0, then either both the equations have real and distinct roots, or both the equations have non-real roots.

D_{1}D_{2} < 0, then one of the equations has real and distinct roots while the other has non-real roots.

D_{1}D_{2} = 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 ax^{2} + 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 x^{2} , then the new roots of the quadratic equation will be α, -α, β , -β

## 6. The relation between roots and coefficients

Let a_{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} +....+a_{n-1}x + a_{n} = 0, with a≠0 be a polynomial equation. Let the roots of the equation be α_{1}, α_{2} , ...,α_{n} .

Sum of the roots taken one at a time:α_{1}+ α_{2} + ...+ α_{n} = - a_{1 }/ a_{0}

Sum of the roots taken two at a time:α_{1}α_{2} + α_{2}α_{3 }+........+α_{1}α_{n }+α_{2}α_{3 }+...+α_{n-1}α_{n }= a_{2} / a_{0}

Sum of the roots taken three at a time:α_{1}α_{2}α_{3} + α_{1}α_{2}α_{4} + ...+α_{n-2}α_{n-1}α_{n} = - a_{3} / a_{0}

and so on

Sum of the roots taken n at a time= (-1)^{n-1}a_{n-1} / a_{0}

Product of the roots , α_{1}α_{2}α_{3} ..α_{n-2}α_{n-1}α_{n }= (-1)^{n}a_{n} / a_{0}

## 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 n_{1}, n_{2} ,....,n_{k} ,m_{1}, m_{2},.....m_{p} are all natural numbers and a_{1},a_{2},.......a_{k}, b_{1},b_{2},...b_{p} 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 -a_{k})^{nk} . If n_{k} is an odd integer then x=a_{k} is called a single point. The function changes the sign on either side of a_{k}

**d)** **Double Point**: Consider (x -a_{p})^{np} . If n_{p }is even integer then x=a_{p} is called a double point. The function has the same sign on either side of a_{p }

Here a_{1} ,a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7} are function zeroes

a_{1} ,a_{2}, a_{4}, a_{5}, a_{6}_{ }are single points because the function changes the sign.

a_{3 }and a_{7 }are double points.

## 9. Quadratic Expressions

__Case 1: a>0__

(i) D<0

(ii) D=0

(iii) D>0

__Case 2: a<0__

(i) D<0

(ii) D=0

(iii) D>0

## 10. Sign of quadratic expression

Let f(x)=ax^{2} + 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 (k_{1},k_{2}) if D≥0, k_{1 }< -b/2a < k_{2} and af(k_{2})>0

f) Exactly one root will lie in (k_{1},k_{2}) if f(k_{1})·f(k_{2})<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)=ax^{2} + 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,

Team Goprep

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Neha Bagri