Maths Notes (Class 10)

CBSE Class 10 Maths Chapter 4 Quadratic Equation

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Quadratic Equation
Quadratic Equation

Quadratic Equation

Quadratic Equation

 
A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0.  E.g.: 2x2 – 3x + 7 = 0,
Application:
  1. Used to find effective resistance of a circuit
  2. Used in the field of communications
  3. Used to find the field of architecture
  4. Used in the field of finance to find demand supply relation
  5. Used to find the projectile of ball throw or bomb throw
  6. Used to find speed of train, boat etc
It is believed that Babylonians were the first to solve quadratic equations. Greek mathematician Euclid developed a geometrical approach for finding solutions of quadratic equations.
Solving of quadratic equations, in general form, is often credited to ancient Indian mathematicians.
Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation.
But when we write the terms of p(x) in descending order of their degrees, then we get the standard form of the equation.
That is, ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.
 

Convert Statement to Quadratic Equation

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
Solution: Let john has x marbles, then Jivanti has 45-x marbles, since total number of marbles is 45.
Now both of them lost 5 marbles each, thus now marble count is
  1. John : x-5
  2. Jivanti : 45-x -5 or 40-x
Given that product of new marble count is 124, that is Marbles with john * Marbles with Jivanti = 124.
(x-5) * (40-x) = 124
Or   x2 – 45x + 324 = 0   , where x is the count of marbles with John.
Numerical 1 : Check if  x(x + 1) + 9 = (x + 2) (x – 2)  is a quadratic equation
Solution: On expanding the equation, we get
x2 + x + 9 = x + 2x -2x -4  
x + 13 = 0.
Since, it is not of the form ax2 + bx + c = 0, it is not a quadratic equation.
 

Solution of a Quadratic Equation by Factorization

For a quadratic equation ax2 + bx + c = 0, a ≠ 0
if  aα2 + bα + c = 0. We also say that x = α is a solution of the quadratic equation
if  aβ2 + bβ + c = 0. We also say that x = αis a solution of the quadratic equation.
Thus if α & β are solution of the quadratic equation ax2 + bx + c = 0. Then we can write
ax2 + bx + c = 0 = (x- α) (x-β), where α & β are solution of this quadratic equation.
In case of factorization method, we re-write the quadratic equation in the form (x- α) (x-β).
Example:  Find solution for quadratic equation x2 – 5x + 6 =0
By splitting the middle term we can rewrite the equation as
x2 – 3x -2x + 6 =0                              as (-3x * -2x = x2  * 6)
Or x(x-3) -2 (x-3) 0
Or (x-3)(x-2) = 0
Or x=3 or x= 2 are solutions for this quadratic equation.
 

Solution of a Quadratic Equation by Completing the Square

In this case, we try to reduce a quadratic equation in the form (px+q)2 = n2
This x = (+n –q)/p     or     (-n –q)/p
For Example, lets solve quadratic equation x2 + 4x – 5=0
We can rewrite this equation as
(x2 + 4x + 4) -4 – 5=0
or (x+2)2 = 32
or X = +3 -2    or  -3-2
or x =1 or -5
 

Nature of Roots

A quadratic equation ax2 + bx + c = 0 has
(i) two distinct real roots, if b2 – 4ac > 0,
(ii) two equal real roots, if b2 – 4ac = 0,
(iii) no real roots, if b2 – 4ac < 0.

Since b2 – 4ac determines whether the quadratic equation ax2 + bx = 0 has real roots or not, b2 – 4ac is called the discriminant of this quadratic equation.

Also Read :

MATHS Revision Notes

Chapter:01  Real Numbers System
Chapter:02  Polynomials
Chapter:03  Pair of Linear Equations in Two Variables
Chapter:04  Quadratic Equation
Chapter:05  Arithemetic Progressions
Chapter:06  Triangles
Chapter:07  Coordinate Geometry
Chapter:08  Introduction to Trignometry
Chapter:09  Some Application Of Trignometry
Chapter:10  Circles
Chapter:11  Constructions
Chapter:12  Area Related to Cirles
Chapter:13  Surface Area Volume
Chapter:14  Stastistics
Chapter:15  Probability

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