Pls. explain completing the square method.

Asked by varshini | 4th Jan, 2014, 09:05: AM

Expert Answer:

Finding solution of quadratic equations by method of Completing Squares

Consider quadratic equations of the form x2 - 36 = 0 and (x+4)2 - 25 = 0.

In both these equations term containing x is completely inside a square. Such quadratic equations can be solved by taking the square roots on both sides as follows:

x2 = 36 => x  = 6 i.e., x = 6 or x = -6.

(x+4)2 - 25 = 0

=> (x+4)2 = 25

=> (x+4) = +25 or -25

=> x = 1 or x = -9

Thus, it can be observed that a quadratic equation involving the x term inside a square can be solved without factorisation.

In fact every quadratic equation ax2 + bx + c = 0 can be reduced to the form (x+a)2 - b2 = 0.

Let us see how, with the help of an example. 

Consider the quadratic equation x2 + 12x + 35 = 0.

x2 + 12x = x2 + 2. 6. x

              = x2 + 2. 6. x + 62 - 62

              = (x + 6)2 - 62            [Using the identity (a + b)2 = a2 + 2ab + b2]

So, x2 + 12x + 35 = (x + 6)2 - 62 + 35

                             = (x + 6)2 - 1

 x2 + 12x + 35 = 0

(x + 6)2 - 1 = 0

(x + 6)2 = 1

Thus, we have reduced the given quadratic equation in the desired form.

Now, taking square root on both sides, we get,

x + 6 = 1 or x + 6 = -1

x = 1 - 6 = -5 or x = -1 - 6 = -7

Thus, -5 and -7 are the roots of the quadratic equation x2 + 12x + 35 = 0.

 

This method of solving a quadratic equation is known as method of completing the squares.

 

The steps involved in solving a quadratic equation by completing the square, are as follows:

Step 1: Make the coefficient of x2 unity.

Step 2: Express the coefficient of x in the form 2  x  p.

Step 3: Add and subtract the square of p.

Step 4: Use the square identity (a + b)2 or (a - b)2 to obtain the quadratic equation in the required form (x+a)2 - b2 =0. Then, take the constant term to the other side of the equation.

Step 5: Take the square root of the obtained equation to get the roots of the given quadratic equation.

Answered by  | 4th Jan, 2014, 11:21: PM

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