the square and the quadratic formula?

Question

Give an example using either completing the square or the quadratic formula and explain each step as if you were teaching someone who had never used the method before.

Answer 1

Given ax^2 + bx + c = 0, find x.

First divide through by a, because the coefficient of the leading term must be 1.

x^2 + (b/a)x + c/a = 0/a = 0

Now move the constant, c/a, to the right side of the equation. We get:

x^2 + (b/a)x = 0 - c/a. = -c/a.

Now we complete the trinomial square on the left side of the equation by taking half the coefficient of the second term and squaring it. Then we add this to both sides of the equation in order to keep it balanced. We do this so that later we can take the square root of both sides of the equation

x^2 + (b/a)x + **(b/2a)^2 = -c/a + (b/2a)^2
** Note that b/2a is equal to 1/2 x b/a

x^2 + (b/a)x + b^2/4a^2 = -c/a + b^2/4a^2

On the right side of this last equation we find a common denominator and perform the necessary multiplication on the appropriate term to get it in the needed format. In this case, we want to make the common denominator 4a^2. So we multiply -c/a by 4a on both the top and bottom to obtain -4ac/4a^2. Our new equation then looks like this:

x^2 + (b/a)x + b^2/4a^2 = -4ac/4a^2 + b^2/4a^2.

Now we can add the numerators in both fractions on the right and commute their order, since it it more natural to subtract a number from another, to obtain this equation:

x^2 + (b/a)x + b^2/4a^2 = (b^2 - 4ac)/4a^2.

Now we begin to find x by extracting the square root of both sides of the equation. To do this, we first express the left side of the equation as the square of the sum of two numbers. After extracting the square roots of both sides, we transpose the constant, b/2a, to the right side of the equation in order to isolate x on the left side of the equation.

(x + b/2a)^2 = (b^2 - 4ac)/4a^2.
(x + b/2a) = [sq rt (b^2 - 4ac)]/2a.
x = - b/2a + or - sq rt (b^2 - 4ac)/2a.
x = [-b + or - sq rt (b^2 - 4ac)]/2a.

Note that we can take the positive or negative square root of b^2 - 4ac because they both equal the same number when squared.

We have just derived the quadratic formula using the method of completing the trinomial square. I hope I have been clear and detailed enough in my explanations.

Answer 2

This is kind of hard to explain online, since most of the symbols we'd use are not available on this online text editor. however, I can refer you to some sites they explain completing the square..
http://www.webmath.com/polycs.html
http://www.purplemath.com/modules/sqrquad.htm
http://216.109.125.130/search/cache?ei=UTF-8&fr=slv8-dyc&p=how+to+complete+the+square&u=darkwing.uoregon.edu/%7Ecphan/math111fall04/completesquare.pdf&w=complete+square&d=GFuJi0VuN2F_&icp=1&.intl=us
As for the quadratic formula, it is derived by completing the square in the general case. Again, it's hard to write out here, so here's a link:
http://www.purplemath.com/modules/quadform.htm
You'd simply substitute the numbers into the formula, as shown, and that's the answer.

Answer 3

I've memorized this one from a cheesy song I learned in middle school. Here it goes: -b, plus or minus radical, b squared minus 4 AC all over two A.

Here it is illustrated if you didn't get it http://id.mind.net/~zona/mmts/miscellaneousMath/quadraticFormula/qesol15.gif