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Free Homework Math Help: Completing The Square

 

Completing The Square

Sometimes, a quadratic equation does not conform to simple factoring methods. Another method that can be used is called completing the square. This method is used when the form ax2 + bx = c exists but cannot be factored. The solution involves using two rules:

The Zero or Additive Identity:        a + 0 = 0 + a = a
The Perfect Square Trinomial:        a2 + 2ab + b2 = (a+b)2

The zero identity is used to put the quadratic equation in a form that can be used by the Perfect Square Trinomial. The Additive Identity is slightly modified to get the form that will be used.

a = 0 = 0 + a → 0 + 0 + a - a → 0 = a - a

By using this property, the equation remains balanced when setting up the Perfect Square Trinomial. In the case of ax2+bx = c, the first step is to divide both sides of the equation by the leading coefficient:

x2   +   b
a		x   =   c
a

Next, the value to be found is one half of the middle coefficient squared. That is half of b/a squared:

(b
2a		)2

This value is added to both sides of the equation:

x2   +   b
a		x   +   (b
2a		)2   =   c
a		   +   (b
2a		)2

This is the Additive Identity equivalent to:

x2   +   b
a		x   +   (b
2a		)2   -   (b
2a		)2   =   c
a

Note that for the quadratic equation to be in proper form for the perfect square, the positive term of the identity has to be next to the rest of the equation containing the variable. The next step involves applying the Perfect Square Trinomial formula

x2 + 2xy + y2 = (x+y)2

to the quadratic equation:

x2   +   b
a		x   +   (b
2a		)2   =   c
a		   +   (b
2a		)2

The result is:

(x   +   b
2a		)2   =   c
a		   +   (b
2a		)2
Take the square root of both sides then subtract (b
2a		)2 from both sides. In this form, the x is by itself and the equation is called the Quadratic Formula.

The quadratic formula results from completing the square

Complete the square example:

3x2 + 6x = 12 a = 2, b = 6

Divide both sides by three:

x2 + 2x = 4

Find    (b
2a		)2   :


(b
2a		)2   =   (6
2*3		)2 = 1

Add one to both sides:

x2 + 2x + 1 = 5

Apply the Perfect Square Trinomial formula:

(x+1)2 = 5

Take the square root of both sides:

x+1 = √5

Solve for x:

x = -1±√5

Using Geometry to Complete the Square

The following example turns a rectangle into a square and subtracts the proper amount of area to balance the result.

Completing the square is an important skill in algebra
Any quadratic equation can be solved by completing the square
Completing the square uses the additive identity

Method for Completing the Square

The steps for completing the square are:

  1. Get the variables on one side of the equation. The left side is preferred by most.
  2. Make sure the equation is in the form ax2 + bx = c.
  3. Divide both sides by the leading coefficient.
  4. Complete the square by adding the square of one half of the middle coefficient to both sides. The middle coefficient is b in ax2 + bx + c which is also the first degree coefficient.
  5. Put the equation in Perfect Square form.
  6. Take the square root of both sides.
  7. Solve for x.