George P. answered • 04/05/13

Penn State Instructor exp.: Algebra, Trig., PreCalculus, and Calculus

Your question may be to complete the square to solve for x?

If that is the case, here is how you would go about doing that:

To complete the square on the left side of the equation, you want

to get a form of (x + a)^{2} on the left side. To do that, you want to

get a form of x^{2} + 2ax + a^{2} because (x + a)^{2} = (x+a)(x+a)= x^{2} + 2ax + a^{2}

This is a perfect square trinomial, which is the square of a binomial.

So, the object is to add a term to the left to get the form of x^{2} + 2ax + a^{2 }

So, here in this example, -12x is the 2ax that you want. Since 2a = -12,

a = -6, Then you want to square this to get a^{2} = 36. This is the constant

that you want to add to get the perfect square trinomial. And since you are

adding this to one side of the equation, you must also add 36 to the other side.

For a more mechanical way of completing the square (assuming the coefficient

is 1 on x^{2} as in this example), you can just divide the coefficient of the x-term by 2,

and then square that result. In this example, divide -12 by 2 to get -6, then square

to get 36. (If the coefficient on x^{2} is not 1, you can divide both sides of the equation

by that coefficient, and then use the method above again.)

Now, to solve this particular problem:

As I mentioned above, we want to add 36 to both sides to complete the square:

x^{2} - 12x +36 = 14 + 36

(x - 6)^{2} = 50 (factoring the left side gives us the binomial squared)

Then, take the square root of both sides to solve for x:

x - 6 = ±√50

x - 6 = ±5√2 (since 50 = 25*2 and √25 = 5)

x - 6 = 5√2 or x - 6 = -5√2

so x = 6 ± 5√2 (Or, you can write the 2 solutions separately as 6 + 5√2 and 6 - 5√2)