Quadratics: Standard Form to Vertex Form

Q:  Write the equation of y = -x2 + 2x + 2 in the form of y = a(x-h)2 + k

A:  To do this, we are going to use a strategy called “completing the square” — a fairly complicated algebra 2 concept.

We are currently in the form y = ax2+bx+c and we want to get to y = a(x-h)2 + k

First, let’s establish a, b, and c in our equation: y = -x2 + 2x + 2

a = -1, b = 2, c = 2

I will walk you through the steps on how to do the problem.  At the end, I will provide some explanation behind the concept and the “why”.

Step 1)  Divide the entire equation by “a”

So, divide everything by -1:

y = -x2 + 2x + 2

y/ -1 = (-x2 + 2x + 2) / -1

And simplify to get:

-y = x2 – 2x – 2

Step 2)  “Complete the Square”

This step involves finding what number to add (or subtract) into the equation that will make the x terms factor nicely.  To find this number, we follow a simple pattern:

Take 1/2 of the coefficient in front of the x term and then square it.

The coefficient in front of x is -2.

1/2 * -2 = -1

Square -1 to get +1.

The number we need to “Complete the Square” is +1.

Add this number to both sides of the equation (remember to do it to BOTH SIDES to keep the equation balanced).  Also, add in by the x’s just for ease, like so:

-y + 1 = x2 – 2x + 1 – 2

So, our modified equation is:

-y + 1 = x2 – 2x + 1 – 2

Step 3)  Factor the x terms:

We are now going to factor the part of the equation I’ve highlighted.  The part we factor involves the x’s and the “complete the square” number that we added to the equation:

-y + 1 = x2 – 2x + 1 – 2

Factoring x2 – 2x + 1 gives (x – 1)(x – 1) OR (x – 1)2

So, we now have:

-y + 1 = (x – 1)2 – 2

Step 4) Isolate y

We have finished the hardest part (completing the square)!  Now, we just solve for y and we are done:

-y + 1 = (x – 1)2 – 2

Subtract 1 from both sides:

-y = (x – 1)2 – 3

Multiply everything by -1:

y = -(x – 1)2 + 3

And there it is!  We have taken the original equation: y = -x2 + 2x + 2 and re-written it in a different form: y = -(x – 1)2 + 3


Some logic behind the process:

Completing the square is the process of figuring out “what number” is needed to add (or subtract) in the equation so that it will factor easily into something like (x-h)2.  Once we determine the number needed, we add it to both sides of the equation to maintain balance.  Remember, we aren’t altering the actual equation, we are just changing its appearance.  Once we found the correct number, the equation will factor the way we need it to.

 

Solving a quadratic

Q:  Solve for n:

4n2 + 3 = 7n
 

Answer:  Since there is an n-squared term, this is a quadratic equation.  In order to solve this, we need to set the whole equation equal to 0 first (so, let’s subtract the 7n over to the left side of the equation):

We get:

4n2 – 7n + 3 = 0  [notice that I put the n’s in order of n-squared, n, and then the constant 3]

Now, there are a few methods you may have learned to can help you solve this:  1)  Factoring or 2) Quadratic Formula or 3) Completing the Square.

This is factorable, so I am going to solve it by factoring.  Keep in mind, if you have never factored something this complicated, you will need to plug it into the quadratic formula (no big deal, just use a = 4, b = -7, c = 3 and plug it in to the big ol’ formula you have).

To factor… set up your parenthesis:

(            )*(              )

The first two terms need to multiply to give 4n2 and the back two terms need to multiply to give + 3.  To multiply and get +3 we either need 2 positive numbers or two negative numbers.  Specifically, we need -1 * -3 or 1*3.

Since we have a -7n in the middle, the back two terms both need to be negative.  Therefore, the two back terms must be -1 and -3:

( ___ – 1 )*( ___ – 3)

Now we need to figure out the front terms.  They must multiply to give 4n2 .  This leaves us with either 4n*1 or 1*4n or 2n*2n.

So you can try each combination in place, distribute it out by hand or visually to see which one will “work”.  Factoring does involve a decent amount of guessing and checking if you want to be good at it.

Through a little guessing and checking, I get:

4n2 – 7n + 3 = (n – 1)*(4n – 3) = 0

Which means either:

n – 1 = 0  or  4n – 3 = 0

Solve for n in each case to get:

n = 1 or n = 3/4

You will get the same answers if you had used the quadratic equation to solve (which may be easier if you are not comfortable with factoring more complicated expressions)