Question

4x^2+6x+3=0 solve by completing the square method

Answer 1

Answer:

x = -3/4 + √ 21/16

or

x = -3/4 - √ 21/16

Step-by-step explanation:

Solving 4x2+6x-3 = 0 by Completing The Square .

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

x2+(3/2)x-(3/4) = 0

Add 3/4 to both side of the equation :

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

Now the clever bit: Take the coefficient of x , which is 3/2 , divide by two, giving 3/4 , and finally square it giving 9/16

Add 9/16 to both sides of the equation :

On the right hand side we have :

3/4 + 9/16

The common denominator of the two fractions is 16 Adding (12/16)+(9/16) gives 21/16

So adding to both sides we finally get :

x2+(3/2)x+(9/16) = 21/16

Adding 9/16 has completed the left hand side into a perfect square :

x2+(3/2)x+(9/16) =

(x+(3/4)) • (x+(3/4)) =

(x+(3/4))2

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

x2+(3/2)x+(9/16) = 21/16 and

x2+(3/2)x+(9/16) = (x+(3/4))2

then, according to the law of transitivity,

(x+(3/4))2 = 21/16

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+(3/4))2 is

(x+(3/4))2/2 =

(x+(3/4))1 =

x+(3/4)

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

x+(3/4) = √ 21/16

Subtract 3/4 from both sides to obtain:

x = -3/4 + √ 21/16

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

x2 + (3/2)x - (3/4) = 0

has two solutions:

x = -3/4 + √ 21/16

or

x = -3/4 - √ 21/16

Note that √ 21/16 can be written as

√ 21 / √ 16 which is √ 21 / 4

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