SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE METHOD

Solving Quadratic Equations by Completing the Square Method :

Here we are going to see how we use the method completing the square to solve a quadratic equation.

Step 1 :

Write the quadratic equation in general form ax2 +bx +c = 0 .

Step 2 :

Divide both sides of the equation by the coefficient of x2 if it is not 1.

Step 3 :

Shift the constant term to the right hand side.

Step 4 :

Add the square of one-half of the coefficient of x to both sides.

Step 5 :

Write the left hand side as a square and simplify the right hand side.

Step 6 :

Take the square root on both sides and solve for x.

Solving Quadratic Equations by Completing the Square Method - Examples

Question 1 :

Solve the following quadratic equations by completing the square method

(i) 9x2 −12x + 4 = 0

Solution :

Step 1 :

Since the coefficient of x2 is 9 not 1, we have to divide the entire equation by 9.

  x2 - (4/3)x + (4/9)  =  0

Step 2 :

Shift the constant term to R.H.S

  x2 - (4/3)x  =  -4/9

Step 3 :

Half of coefficient of x is -2/3. We have to square this and add it on both sides.

  x2 - (4/3)x + (-2/3)2  =  -4/9 + (-2/3)2

 [x - (2/3)]2  =  (-4/9) + (4/9)

 [x - (2/3)]2  =  0

 [x - (2/3)]2  =  0

x - (2/3)  =  0   and x - (2/3)  =  0

x = 2/3 and x = 2/3.

Hence the solutions are 2/3 and 2/3.

(ii)  (5x + 7)/(x - 1)  =  3x + 2

Solution :

5x + 7  =  (3x + 2)(x - 1)

5x + 7  =  3x2 - 3x + 2x - 2

3x2 - x - 5x - 2 - 7  =  0

3x2 - 6x - 9  =  0

Divide the entire equation by 3.

x2 - 2x - 3  =  0

x2 - 2x  =  3

Half of coefficient of x is -1. Now we are going to add the square of -1 on both sides.So, we get 

x2 - 2x + (-1)2 =  3 + (-1)2

x2 - 2x + 1 =  3 + 1

(x - 1)2  =  4

(x - 1)  =  4

(x - 1)  =  ± 2

x - 1  =  2   and x - 1  =  -2

x  =  2 + 1   and x  =  -2 + 1

x  =  3 and x = -1

Hence the solutions are 3 and -1.

After having gone through the stuff given above, we hope that the students would have understood, "Solving Quadratic Equations by Completing the Square Method". 

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