## Tuesday, November 27, 2012

I was working with a student on completing the square and it tends to be a more tricky method of factoring than using the quadratic formula.  There are several steps to remember.

Recall how we factored and solved quadratic equations using the reverse FOIL method. An example of this type of factoring is x^2 + 5x + 4 = 0 factored is (x +4)(x + 1) = 0. Sometimes it's quite difficult to solve quadratic equations using this method, so we can solve by a method known as completing the square.
The idea behind completing the square is to turn a binomial into a perfect square trinomial. For example, consider the binomial x2 + 6x. The perfect square trinomial with the first two terms x2 + 6x is x2 + 6x + 9 because (x + 3)(x + 3) = x^2 + 6x + 9. Notice how we added a 9 to x^2 + 6x. The question we ask ourselves is, “What number squared equals 9?” We know that 3^2 = 9. Also notice that 3 is half of the coefficient of the middle term 6x. We take half of the middle term, square it and
add it to form the perfect square trinomial. Then we factor the trinomial.

Examples: Complete the square and factor the perfect square trinomial.
1. x^2 + 8x

Step one: Take half the coefficient of the middle term. (1/2)(8) = 4.
Step two: Square the result in step one. 4^2 = 16.
Step three: Add the result in step two to the binomial to form the trinomial x^2 + 8x + 16.
Step four: Factor the trinomial. (x + 4)(x + 4) or (x + 4)^2.

x^2 – 7x
Step one: Take half of the coefficient of the middle term. (1/2)(-7) = -7/2.
Step two: Square the result in step one. (-7/2)^2 = 49/4.
Step three: Add the result in step two to the binomial to form the trinomial x^2 – 7x + 49/4.
Step four: Factor the trinomial. (x – 7/2)(x – 7/2) or (x -7/2)^2

To solve quadratic equations by completing the square we must remember the following:
1. Make sure the coefficient of the squared term is one. If it is not one, we must divide both sides of the equation by the coefficient of that term. For example, if the term is 2x2, we must divide both sides of the equation by 2.
2. Get all variables on one side of the equation and the constants on the other side. This makes sure we have a binomial in the form x2 + bx.
3. Take half of the coefficient of the middle term, square it and add it to both sides of the equation.
4. Factor the perfect square trinomial.
5. Solve the equation using the square root property and check answers by substituting into the original equation.

Note if the coefficient in front of the x^2 term is not 1, must divide the equation by the coefficient before completing the score.