Developing Quadratic Formula
Developing the quadratic formula is a natural combination in the quest to solve quadratic equations.
We first solve the quadratic equation by factoring in using a zero principle.
The weakness with this method is that not all quadratic equations can be factored.
So we needed a better method especially one that would always work.
Mathematicians developed completing the square it always works, but can be more difficult to use than we would like.
To solve the difficulty issue. Why not solve the general quadratic and just use a formula.
Developing the quadratic formula we start with the general quadratic equation and the quadratic formula.
The if-then statement below shows the quadratic formula and the general quadratic
equation, so if AX squared plus BX plus C equals 0
Then we know for sure
That X equals a negative b plus or minus the square root of b squared minus 4ac all over 2a
so we'll have our X as long as we know what a B and C are and
We can find the a B and C from the general quadratic
equation.
We will now show how this formula developed notice that the general quadratic equation is set equal to zero.
Developing Quadratic Formula
ax squared plus bx plus C equals zero and we can divide both sides by a
And we do that so that we can get rid of the numerical coefficient on the x squared term.
So we get X squared plus B over a X plus C over a equals zero
Now watch each step and decide. Can you see what we did and how was done?
The algebra is a little general has some fractions in it
But it's easy to understand if you have a pretty good working knowledge of algebra.
So now let's see. What do we want to do next when we were completing the square. That's right
Get rid of the C over a so we're going to subtract C over a from both sides
And we get X squared plus B over ax plus I left a little blank there
equals a negative C over A
So it did disappear on the left and reappeared on the right as a negative by subtracting from both sides.
Now, what do we want to do next in completing the square we wanted to know and what we could put in the blank that.
Would make that a perfect square trinomial and then whatever that was. We're going to add it onto the right hand side.
Now the way we did that was we took half the middle term
B over a and B B over 2a and we square B over 2a
And we get B squared over 4a squared
and we have to add that on to both sides to keep our
equation balanced
Now if we look at this, we want to factor the left-hand side.
Down to something times itself and on the right-hand side
We want to be able to find a common denominator between those two fractions
so let's do both of those and then we'll talk about.
So we factored the left into X plus B over 2a times the quantity X plus B over 2a
On the right we found the common denominator or 4a squared by multiplying the numerator denominator
by 4a
So now we'll rewrite the left-hand side as a
Quantity squared and we'll add the two fractions together on the right putting the positive 1 first
So we get the quantity X plus B over 2a squared
Equals b squared minus 4ac all over 4a squared.
Now look at what we did
We added the ones on the right we put the B squared first is so it's kind nicer to have that
B squared minus 4ac on top rather than the other way around
And we put them over the results over the common denominator on the left. We just rewrote what was
In there when we had something times itself so you could write it as a squared
Now we're going to take the square root of both sides
Okay, now when we take the square root of the left-hand side
The square root. It's going to get rid of the square and the right-hand side. We can't take the square root of the numerator
But in this case the denominator is a perfect square so we can take the square root of 4a squared
And we get X plus B over 2a on the Left
Square root cover to the squared and on the right we couldn't take the square root of the numerator,
But we go to the denominator. So we did that now if you look we only have it one more thing by the X
So all we gotta do now is subtract B over 2a from both sides
So the X now is by itself and we have a negative B over 2a
Plus the square root of b squared minus 4ac all over 2a
Now fortunately both these have a common denominator
So all we have to do is combine
their numerators
So we get x equals b squared plus or minus the square root of b squared minus
4ac all over 2a and that is the quadratic formula
So that's pretty nice. So we finally
Solve the general quadratic. So now we can just use the formula
all we need to know is the a B and C and be little careful with our
Order of operations in solving that quadratic
formula.
I'd like to just flick jump at what happened between row 3 and Row 4
Going from Row 3 is a single equation to Row 4, which is a single equation
The thing on the left. I'm more interested in we factored that and so if you look at those factors are
Those the factors of that
the fourth row factors
Will they multiply out and give you the third row on the left? So I'm going to kind of review that
So checking the factoring going from Row 3 to Row 4
we're going to take that X plus B over 2a and
multiply it times X plus B over 2a and so this is in Row 4 and we're hoping to get the
Trinomial in Row 3. So let's see. We just use the distributive property
x times B over
x times X is x squared
x times B
over 2a is X times B over 2a
Plus B over 2a times X is B over 2a plus X and then last is B over 2a times B over 2a?
Is B squared over 4a squared?
Now the next step we factor out the X out of the middle two terms so we can combine their numerical coefficients
So then we can add that B over 2a plus B over 2a and we get 2 B over 2a
Then in the last row we cancel the twos so we get x squared plus B over ax
Equals B squared over 4a squared and if you look and go back from Row 4 to 3
That's exactly what we're supposed to get
So we have developed a quadratic formula I
think this is a very interesting proof because
Quadratic formula is kind of complicated and you wonder where'd it come from?
This is where it came from
And all the algebra is straightforward we had to deal with little fractional
Algebra, and we also had to deal with some factoring but works out real nice.
