Quadratic equation Chapter 4 Class 10
Basic concepts on Quadratic Equation class 10, chapter 4, Nature of roots, Discriminant, Quadratic Formula, method of completing the square. Complete explanation of quadratic equations
QUADRATIC EQUATION :
An equation whose degree is 2 is called a quadratic equation.
General Quadratic Equation is ax^{2} + bx + c = 0
Here "a" is the coefficient of x^{2} ,
"b" is the coefficient of x and
"c" is the constant term.
Difference between the quadratic equations and quadratic polynomials.
Quadratic equations are very similar to the quadratic polynomials. But they are different from each other because of the following reasons.
Quadratic Equations

Quadratic Polynomials

General Quadratic Equations is
ax^{2} + bx + c = 0

General Quadratic Polynomial is
P(x) = ax^{2} + bx + c

Solutions of quadratic equations are called its roots.

Solutions of quadratic polynomials are called its zeroes.

Methods of Solving Quadratic Equations
There are mainly three methods of solving Quadratic equations
1) Factor Method
2) By Quadratic Formula
3) My the method of Completing the Square.
ROOTS:
Solutions of the quadratic equations are called its roots. A quadratic equation have two roots.
RELATION BETWEEN ROOTS AND COEFFICIENTS:
Quadratic Equation From The Roots:
x^{2} – (Sum of roots)x + Product of roots = 0
Nature Of The Roots Of The Quadratic Equation:
General quadratic equation is
ax^{2} + bx + c = 0
Discriminant of quadratic equation is denoted by D and is given by
D = b^{2}  4ac
If D > 0 then roots are real and unequal or distinct or different)
If D = 0 then roots are real and equal
If D < 0 then roots are not real
If D ≥ 0 then roots are real
Quadratic Formula For Solving The Quadratic Equations:
Method of completing the square:
Write the given quadratic equation
ax^{2} + bx + c = 0
Make the coefficient of x^{2} unity
Bring the constant term to the right hand side.
Taking the square root on both side
Example
Find the roots of
5x^{2}  6x  2 = 0 by the method of completing the square
Given equation is
5x^{2}  6x  2 = 0
D = b^{2}  4ac
⇒ D = (6)^{2}  4 x 5 x 2 = 36+40 = 76 > 0
⇒ Roots are real and distinct
⇒ Roots of Q. E. are exists
Now we apply method of completing the square as follows
Step (1) Make the coefficient of x^{2}
unity ( divide the equation by 5)
Step (2) Bring the constant term to the RHS
Step (3)
Step (4) Completing
the square and find the square root

Method of solving the Q. E. by using quadratic formula
Find the roots of 5x^{2}  6x  2 = 0 by the Quadratic Formula Given equation is 5x^{2}  6x  2 = 0 D = b2  4ac ⇒ D = (6)^{2}  4 x 5 x 2 = 36 + 40 = 76 > 0 ⇒ Roots are real and distinct ⇒ Roots of Q. E. are exists Now we apply the quadratic formula

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