## Friday, May 24, 2013

Now we will see the important problems of quadratic equations to be solved using the perfect square method:

Few steps to be followed to solve the quadratic equation using the perfect square method:

1) Shift the constant term to the right-hand side of the equation.
2) Divide both sides of the equation by the coefficient of 2.
3) Find the third term of the left-hand side to make it as a perfect square.
4) Use the formula " Third Term = ( 1/2 coefficient of x ) 2.
5) Add this third term so obtained as mentioned in step 4 both sides of the equation.

a]  Solve the quadratic equation  x 2 - 18 x  + 65 = 0 using perfect square method.

Solution:
1) Shift 65 to RHS
2)  2 - 18 x  = - 65                  Here third term = ( 1/2 coefficient of  x ) 2
= ( 1/2 (18) 2
= ( 2
= 81
2 - 18 x  + 81 = - 65 + 81
(  - 9 ) 2 = 16
(  - 9 )  = + 4 or  - 9 )  = - 4
= 9 + 4 or   =  9 - 4
= 13 or   =  5
3)  So the roots of the equation are 5 or 13 so Solution Set = { 5, 13 }

b]  Solve the quadratic equation  x 2 - 5 x  + 6 = 0 using perfect square method.

Solution:
1) Shift 6 to RHS
2)  2 - 5 x  = - 6                  Here third term = ( 1/2 coefficient of  x ) 2
= ( 1/2 (5) 2
= ( 5/2 2
= 25/4
2 - 5 x  + 25/4 = - 6 + 25/4
(  - 5/2 ) 2 = (25-24)/4
(  - 5/2 ) 2 = 1/4
(  - 5/2 )  = + 1/2 or  - 5/2 )  = - 1/2
= 5/2 + 1/2 or   =  5/2 - 1/2
= 6/2 or   =  4/2
= 3 or   =  2
3)  So the roots of the equation are 2 or 3 so Solution Set = { 2, 3 }

c]  Solve the quadratic equation  x 2 - 6 x  + 2 = 0 using perfect square method.

Solution:
1) Shift 2 to RHS
2)  2 - 6 x  = - 2                  Here third term = ( 1/2 coefficient of  x ) 2
= ( 1/2 (- 6) 2
= ( - 3 2
= 9
2 - 6 x  + 9 = - 2 + 9
(  - 3 ) 2 =  7
(  - 3 )  = + √ 7 or  - 3 )  = - √ 7
= 3 + √ 7 or   =  3 √ 7
3 + √ 7 or   =  3 - √ 7
3)  So the roots of the equation are 3 + √ 7 or 3 - √ 7 so Solution Set = { 3 + √ 7,  3 - √ 7 }

d]  Solve the quadratic equation  x 2 - 5 x  + 2 = 0 using perfect square method.

Solution:
1) Shift 2 to RHS
2)  2 - 5 x  = - 2                  Here third term = ( 1/2 coefficient of  x ) 2
= ( 1/2 (- 5) 2
= ( - 5/2 2
= 25/4
2 - 5 x  + 25/4 = - 2 + 25/4
(  - 5/2 ) 2 =  (- 8 + 25)/4
(  - 5/2 ) 2 =  17/4
(  - 5/2 )  = + (√ 17)/2 or  - 5/2 )  = -  (√ 17)/2
= 5/2 + (√ 17)/2 or   =  5/2 - (17)/2
= (5 + √ 17)/2 or   =  (5 - √ 17)/2
3)  So the roots are (5 + √ 17)/2 or (5 - √ 17)/2 so Solution Set = {(5 + √ 17)/2,  (5 + √ 17)/2 }