Question

solve the following quadratic equation (real roots only) by the method of completing the perfect square
$4 x^2 + 12x + 1 = 0$
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Answer 1

3.2 Solving 4x2-12x+1 = 0 by Completing The Square .

Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :

x2-3x+(1/4) = 0

Subtract 1/4 from both side of the equation :

x2-3x = -1/4

Now the clever bit: Take the coefficient of x , which is 3 , divide by two, giving 3/2 , and finally square it giving 9/4

Add 9/4 to both sides of the equation :

On the right hand side we have :

-1/4 + 9/4 The common denominator of the two fractions is 4 Adding (-1/4)+(9/4) gives 8/4

So adding to both sides we finally get :

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

Adding 9/4 has completed the left hand side into a perfect square :

x2-3x+(9/4) =

(x-(3/2)) • (x-(3/2)) =

(x-(3/2))2

Things which are equal to the same thing are also equal to one another. Since

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

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

then, according to the law of transitivity,

(x-(3/2))2 = 2

We'll refer to this Equation as Eq. #3.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(x-(3/2))2 is

(x-(3/2))2/2 =

(x-(3/2))1 =

x-(3/2)

Now, applying the Square Root Principle to Eq. #3.2.1 we get:

x-(3/2) = √ 2

Add 3/2 to both sides to obtain:

x = 3/2 + √ 2

Since a square root has two values, one positive and the other negative

x2 - 3x + (1/4) = 0

has two solutions:

x = 3/2 + √ 2

or

x = 3/2 - √ 2

Hope it is helpful for you