Quadratic equation

Quadratic equation:-An equation of the form ax²+bx+c=0 where a,b and c ∈ R and a≠0 is called a quadratic equation.

Note-(1)An equation of degree 2 is called a quadratic equation.

Ex-1.x²+5x+6=0

      2.x²+7x+10=0

      3.x²-5x+10=0

      4.2x²+8x+6=0

Solution or roots of a quadratic equation:-  If p(x)=0 is a quadratic equation,then the zeros of the polynomial p(x) are called the solutions or roots of the quadratic equation p(x)=0.

Note:-1.Since the degree of a quadratic equation is 2,it has 2 roots or solutions.

2.x=a is the root of p(x)=0,if p(a)=0.

Method of solving a quadratic equation-

(a)spliting the middle term:-

1. x²+5x+6=0.

split the middle term 5x=3x+2x ,because multiplication of extreme terms(x²×6)=6x²=3x×2x.

Here x²+3x+2x+6=0

       ⇒ x(x+3)+2(x+3)=0

        ⇒(x+3)(x+2)=0

         ⇒x=-3 ,-2

2. x²-7x+12=0 

⇒x²-4x-3x+12=0, because multiplication of extreme term x²×12=(-4x)(-3x)=12x²

⇒x(x-4)-3(x-4)=0

⇒(x-4)(x-3)=0

⇒x=4, 3

3. 2x²+12x+16=0

⇒2x²+8x+4x+16=0

⇒2x(x+4)+4(x+4)=0

⇒(x+4)(2x+4)=0

⇒x=-4 ,-2

solve the following quadratic equation.

1.x²+5x+4=0

2.x²-5x+4=0

3.x²+5x-6=0

4.x²-5x-6=0