CAT  >  Quantitative Aptitude (Quant)  >  Quadratic Equations: Notes & Solved Examples

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

1 Crore+ students have signed up on EduRev. Have you?

Polynomials
Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
Definition of a Polynomial

Let Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT be real numbers and x is a real variable. Then f(x) = Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CATQuadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT  is called a real polynomial of real variable x with coefficients Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

Example: Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT is a polynomial

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial.

  • Example: Degree of Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT is 3 as the maximum power of the variable x is 3.
  • On the basis of degree, the polynomials are classified as Linear (Degree 1), Quadratic (degree 2), Cubical (Degree 3), bi-Quadratic (degree 4), and so on.

Question for Quadratic Equations: Notes & Solved Examples
Try yourself:What is the degree of a polynomial of 4x7+9x5+5x2+11?
View Solution

Examples on Degree of a Polynomial

Q.1. Find the degree of the polynomial 6x4+ 3x2+ 5x +19

Ans: The degree of the polynomial is 4. 

Q.2. Find the degree of the polynomial 6x3+4x3+3x+1

Ans: The degree of the polynomial is 3.


Remainder Theorem

If any polynomial f(x) is divided by (x - a) then f(a) is the remainder.

  • Example: f(x) = x2 - 5x + 7 = 0 is divided by x - 2. What is the remainder?
    R = f(2) = 22 - 5 × 2 + 7 = 1.

Factor Theorem

If (x - a) is a factor of f(x), then remainder f(a) = 0. (Or) if f(a) = 0, then (x - a) is a factor of f(x),

  • Example: When f(x) = x2 - 5x + 6 = 0 is divided by x - 2, the remainder f(2) is zero which shows that x - 2 is the factor of f(x)

Examples on Remainder & Factor Theorem

Q.1. Find the remainder by using the remainder theorem when polynomial x3−3x2+x+1 is divided by x−1.

Ans: 

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT


Q.2. Find the remainder when p(x)= −x+3x2−1 is divided by x+1.

Ans: 

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

General Theory of Equations

  • An equation is the form of a polynomial which has been equated to some real value.
  • Example: 2x + 5 = 0, x2 - 2x + 5 = 7, 2x2 - 5x2 + 1 = 2x + 5 etc. are polynomial equations.

Root or Zero of a Polynomial Equation

  • If f (x) = 0 is a polynomial equation and f (α) = 0, then α is called a root or zero of the polynomial equation f(x) = 0.

Linear Equation

Linear Equation with One Variable

A linear equation is 1st degree equation. It has only one root. Its general form is a
Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT


Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

 

Linear Equation with Two Variables

  • Example: 2x + 3y = 0. We need two equations to find the values of x and y.
    If there are n variables in an equation, we need n equations to find the values of the variables uniquely.
  • Sometimes, even the number of equations are equal to the number of variables, we cannot find the values of x and y uniquely.
    For example, 3x + 5y = 6 & 6x +10y = 12
  • These are two equations, but both are one and the same. So different values of x and y satisfy the equation and there is no unique solution. It will has infinite number of solutions.
  • The number of solutions is clearly described below for the set of equations with 2 variables.
    Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
  • These equations can be:
    (i) Inconsistent means have no solution ifQuadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
    (ii) Consistent and has infinitely many solutions if
    Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
    (iii) Consistent and have unique solution if
    Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

Quadratic Equations

Quadratic Equation in “x” is one in which the highest power of “x” is 2. The equation is generally satisfied by two values of “x”.

  • The quadratic form is generally represented by ax2 + bx + c = 0 where a ≠ 0, and a, b, c are constants.
  • For Example:
    x2 - 6x + 4 = 0
    3x2 + 7x - 2 = 0
  • A quadratic equation in one variable has two and only two roots, which are
    Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT 
    Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

Nature of Roots

  • The two roots of any quadratic equation always depend on the value of b2 - 4ac called discriminant (D).
    (i) D > 0 Real and unequal roots
    (ii) D = 0 Real and equal
    (iii) D < 0 Imaginary and unequal
    Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

Question for Quadratic Equations: Notes & Solved Examples
Try yourself:Roots of a quadratic equation are imaginary when discriminant is:
View Solution


Sum and Product of Roots

  • If α and β are the two roots of ax2 + bx + c = 0,
  • Then sum of roots = α + β = Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
  • And product of roots = αβ = Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

Formation of Equation from Roots

  • If α and β are the roots of any quadratic equation then that equation can be written in the form: X2 − (α + β)X + αβ = 0
    i.e. X2 - (sum of the roots) X + Product of the roots = 0

Some Important Results
Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT                     

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
-----------------------------------------------
Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

Maximum and Minimum value of a Quadratic Equation

The quadratic equation ax2 + bx + c = 0 will have maximum or minimum value at x = - b/2a. If a < 0, it has maximum value and if a > 0, it has minimum value.

The maximum or minimum value is given by  Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

Solved Examples on Quadratic Equation

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT


Ex.2. A and B went to a hotel paid Rs. 84 for 3 plates of Idli and 5 plates of Dosa. Whereas B took 5 plates of Idli and 3 plates of Dosa and paid Rs. 76. What is the cost of one plate of Idli.

  • 3I + 5D = 84 ……….(1)
  • 5I + 3D = 76 ………(2)
  • Equation (1) × 3 - equation (2) × 5, we get
    ⇒ 16I = 128
    ⇒ I = 8
  • Each plate of Idli cost Rs. 8.


Ex.3. Find the values of x and y from the equations Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT 

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT


Ex.4. Find x and y from Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
 

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT
Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT


Ex.5. Aman won a competition and so he got some prize money. He gave Rs. 2000 less than the half of prize money to his son and Rs. 1000 more than the two third of the remaining to his daughter. If both they got same amount, what is the prize money Aman got?

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT


Ex.6. How many non negative integer pairs (x, y) satisfy the equation, 3x + 4y = 21?

  • Since x and y are non negative integers. Start from x = 0.
  • If x = 0 or 2, y cannot be integer.
  • For x = 3, y = 3.
    And for x = 7, y = 0.
  • These two pairs only satisfy the given equation.


Ex.7. If (x - 2) is a factor of x3 - 3x2 + px + 4. Find the value of p.

  • Since (x - 2) is a factor, f(2) = 0.
  • ∴ 23 - 3(22) + (2)p + 4 = 0
    ⇒ p = 0


Ex.8. When x3 - 7x2 + 3x - P is divided by x + 3, the remainder is 4, then what is the value of P ?

  • f(- 3) = 4
  • ∴ (- 3)3 - 7(- 3)2 + 3(- 3) - P = 4
  • P = - 103.


Ex.9. If (x - 1) is the HCF of (x3 - px2 + qx - 3) and (x3 - 2x2 + px + 2). What is the value of ‘q’?

  • Since (x - 1) is HCF, it is a factor for both the polynomials.
  • ∴ 13 - p(1)2 + q(1) - 3 = 0          
  • - p + q = 2
  • And 13 - 2(12) + p(1) + 2 = 0
  • p = - 1
  • ∴ q = 1


Ex.10. Find the roots of the quadratic equation x2 - x - 12 = 0.

  • x2 - x - 12 = 0
  • x2 - 4x + 3x - 12 = 0
  • x(x - 4) + 3(x - 4) = 0
  • (x - 4) (x + 3) = 0
  • x = 4 or -3
  • The roots are 4 and -3.
The document Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT is a part of the CAT Course Quantitative Aptitude (Quant).
All you need of CAT at this link: CAT
163 videos|163 docs|131 tests
163 videos|163 docs|131 tests
Download as PDF

How to Prepare for CAT

Read our guide to prepare for CAT which is created by Toppers & the best Teachers

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

pdf

,

Important questions

,

Objective type Questions

,

Previous Year Questions with Solutions

,

shortcuts and tricks

,

study material

,

past year papers

,

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

,

mock tests for examination

,

MCQs

,

Exam

,

video lectures

,

Free

,

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

,

practice quizzes

,

Summary

,

ppt

,

Viva Questions

,

Semester Notes

,

Extra Questions

,

Sample Paper

,

Quadratic Equations: Notes & Solved Examples | Quantitative Aptitude (Quant) - CAT

;