POINT OF INTERESTCopyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to ingrid.stewart@csn.edu.

Create the Quadratic Formula .

Let's solve the

General Formof the quadratic equation forx.This is done using the

Square Root Property, but first we will have to change theGeneral Formto the form .Step 1:

Factor

out of theaGeneral Formas follows:Step 2:

Use the

Perfect Squaresformulaand create a perfect square inside the parentheses.(A + B)(A + B) = A^{2}+ 2AB + B^{2}In our case,

. We must change the middle term toA = x2(, namely to . This makes the last termx)(?)equal to .BTo create a perfect square, we must add

, namely . But because of theB^{2}outside the brackets, we must actually add .aTo preserve the value of we must then subtract again as follows:

We get .

Step 3:

We will now factor the

perfect squaresexpression in the brackets to get the followingPlease note that is a constant term (no

x's).Step 4:

Now let's use the

Square Root Propertyto solve the equation in Step 3 forx.We will move the constant term to the right side of the equal sign as follows:

Next, we will divide both sides by

as follows:awhich can be simplified to .

We will continue to simplify the fraction on the right of the equal sign. Study it carefully! All we did was fraction operations!

Then we can write .

Finally, we are ready to apply the

Square Root Propertyto getNow we will isolate the

xas follows:Finally, we will do some major "clean-up" to arrive at the

Quadratic Formula. Study the square root and fraction manipulations carefully!

and we end up with the

Quadratic Formula.