To solve by completing the square, what value should you add to each side of equation?

Question

x^2+16x=-6

Answer 1

Whenever you have to do this kid of question of completing a square,first make sure that the coefficient of x^2 is 1.. for example if I had the equation 3x^2 + 5x=7 and I had to complete the square

I would first divide the equation by 3 on both sides to make the coefficient of x^2 as 1.

so my equation will then be

x^2 + 5x/3=7/3

next step is to see the coefficient of x which is now 5/2 .

we willl take half of that ie 5/4 .this is the number whose square you have to add on both sides of the equation to convert it into a perfect square.

 

 

so now lets see your question.it is x^2+16x=-6

here as you see the coeff of x^2 is already 1.so you dont need to make it 1.

now you see the coeff of x is 16.half of it is 8 ..so you add 8^2 on both sides.

your equation becomes x^2 +16x+64= -6 +64    ie

(x+8)^2= 58  and you are done:)

 

Answer 2

Sounds like you want to take the guesswork out of completing the square.  So, let's think about that.

 

Once the left-side expression looks like  Ax^2 + Bx + C, we think FOIL (First-Outside-Inside-Last) and realize that in order for this expression to be a square, the factors must look like:

          (Mx + N)(Mx + N)

and

             (M^2)x^2 + 2NMx + N^2   must equal  Ax^2 + Bx + C

so:

            M = SQRT(A)

            N =  B / (2M)  =  B / ((2SQRT(A))

 

To simplify this calculation, the process for completing the square includes the first step of making A=1.  So,

            M = SQRT(1) =  1

            N  =  B / ((2SQRT(1))  =  B/2

 

Once you understand those calculations, you can forget them and complete the square easily.  Often teachers or tutors assume that you know these calculations, so they skip over them.  For example, in this problem, we have x^2 + 16x, so the factors are (x + 16/2) or (x + 8).  Thus, the constant amount (C) on the left side of an equation must be 64 and we also add an equal amount to the right side of the equation (and later take the square root of both sides so we can find x).

 

= = = = = = = = = = =

 

Now, let's do all that at once.

 

Starting with an equation:

     x^2+16x=-6

we already have A=1  (if not, we divide everything by the coefficient of x^2)

then we find  B = 16/2 = 8  so out factors will become (x+8)(x+8)   [note:  sometimes, this may be negative]

and that means adding 8^2 to both the left side and to the right side.

 

Once you understand the process, it is easy.

Answer 3

x² + 16x = -6

 

 

When completing the square, we add a value to one side of the equation so that the expression on that side is a perfect square.  Anything we do to one side of the equation, we do to the other.

 

If we add 64 on both sides of the equation, the left side of the equation will be a trinomial that is a perfect square.

 

x² + 16x + 64 = -6 + 64

 

(x + 8)(x + 8) = -6 + 64

 

(x + 8)² = 58

 

 

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