QUADRATIC EQUATION

If you set the polynomial ax2 +bx + c equal to zero, where a, b and c are constants and

a ≠ 0 , there is a special name for it. It is called a quadratic, You can find the values for x that make the equation true.

Example:     x2 – 3x +2 =0

To find the solutions, also called roots of an equation, start by factoring whenever possible. You can factor x2 – 3x +2 into (x-2)  (x-1) , making the quadratic equation:

(x-2) (x-1) = 0

To find the roots, set each binomial equal to 0. That gives you:-

(x-2) = 0      or     (x-1) =0

Solving for x, you get x=2 or x=1

The solutions to a quadratic in the form ax2+bx+c=0 can also be found using the quadratic formula provided a, b and c are real numbers and a ≠ 0 then:

EXAMPLE:- Find the Solution of 2x2 + px +9 = 0

Solution :- a=2 , b=9 , c=9

=  -9±3 / 2

 

ANS= 3     or -3/2

Sum of the roots of quadratic equation =  -b/a

Product of the roots of a quadratic equation = c/a

 

 

  • D= (b2 – 4ac) is the discriminant of the quadratic equation.

 

If D < 0 (ie. The discriminant is negative) then the equation has no real roots.

If D= 0 then the quadratic equation has two equal roots.

If D> 0 (ie. The discriminant is positive) then the quadratic equation has two distinct roots.

 

Graph of a Quadratic Expression:-

Let f(x)= ax2 + bx +c , Where a, b, c are real and a ≠0, Then y=f(x) represent a parabola, Whose axis is parallel to y- axis. This gives the following cases:

 

  • a> 0 and D (b2 -4ac) < 0 (Roots are imaginary).

F(x) > 0 V x ER

 

  • a > 0 and b = 0 (The roots are real and equal)

f(x) will be positive for all values of x except at the written where f(x) = 0

3)When a> 0 and D > 0 (The roots are real and distinct)

F(x) will be equal to zero when x is equal to either of ⋉or β.

 

F(x) > 0 V x E (-∞ , ⋉ ) U (β, ∞)

AND f(x) < 0 V x E (⋉ , β )

 

4.When a < 0 and D=0 (Roots are imaginary)

f(x) < 0 V x ER

 

 

5.When a<0 and D=0 (Roots are real and equal)

F(x) is negative for all values of x except at the vertex Where f(x)= 0

 

 

 

6.When a<0 and D>0 (Roots are real and equal)

F(x) < 0   V x E ((-∞ , ⋉) U (β, ∞)

F(x) > 0   V x E(⋉ , β )

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