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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)

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