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Thursday, April 20, 2023

find the roots of the following quadratic equations by factorization

we shall study quadratic equations and solve them by various methods. In this topic, we will find the roots of the following quadratic equations by factorization method.

Roots of Quadratic Equation Class 10 

Now we will discuss how to solve the following quadratic equation by factorization method


Example -1 

Find the roots of the quadratic equation x2-9x+18=0 by factorization method.


Solution 

The given equation is  x2-9x+18=0 


    x2-(6+3)x + 18 =0 


    x2-  6x-3x + 18 =0 


    x(x- 6)-3(x - 6) =0 


    (x- 6)(x - 3) =0


    x- 6= 0 or x - 3 =0 


    x= 6    or   x=3 


Hence, x= 6    or   x=3 are the roots of the given quadratic equation.


Example -2 

Find the roots of the quadratic equation 2x2-5x+3=0  by factorization method.


Solution 

The given equation is  2x2-5x+3=0 


  2x2 - (2+3)x+3=0


  2x2 - 2x-3x+3=0


    2x(x-1)-3(x - 1)=0


    (x-1)(2x - 3)=0 


    x-1=0    or   2x - 3=0 


     x  =1  or x = 32 


Hence, x  =1  or x = 32  are the toots of given quadratic equation.


Example -3 

Find the roots of the quadratic equation 6x2-x - 2 =0  by factorization method.


Solution 

The given equation is  6x2-x - 2 =0  


   6x2 -(4 - 3)x   - 2 =0 


   6x2 -4x  + 3x   - 2 =0


   2x(3x - 2)+  1(3x  - 2) =0 


   (3x - 2)+ (2x  +1 ) =0 


   3x - 2=0    or  2x  +1  =0 


⟹   x=23      or  x = -12    


Hence,  x=23      or  x = -12   are the roots of given quadratic equation.



Example -4

Find the roots of the quadratic equation 3x2-26x+2=0 by factorization method.


Solution

 The given equation is 3x2-26x+2=0  

⟹    3x2-(6+6)x+2=0               



 ⟹    3x2-6x+6x+2=0 



 

⟹  3x(3x-2)-2(3x-2)=0

  

(3x-2)(3x-2)=0


3x-2=0  or 3x-2=0

 

⟹      x=23   or        x=23

  

Hence,   x=23   or     x=23     are the roots of the given quadratic equation.


 Example -5 

Find the roots of the quadratic equation 15a2x2-16abx-15b2=0   by factorization method.


Solution 

The given equation is   15a2x2-16abx-15b2=0 


⟹  15a2x2-(25ab-9ab)x-15b2=0



⟹  15a2x2-25abx+9abx-15b2=0 


⟹  5ax(3ax-5b)+3b(3ax-5b)=0


⟹  (3ax-5b)(5ax+3b)=0 


⟹      3ax-5b=0or5ax+3b=0 


⟹      x=5b3a     or x=-3b5a=0 


Hence,  x=5b3a     or x=-3b5a=0 

 are the roots of the given quadratic equation.


Now you must have known the method of finding roots of quadratic equation by factoring.


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