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 )