Quadratic Formula

Hi guys! .. I'm Sandy and I'm your scribe.

Today we continued from yesterdays lesson and we started with this question.

1. a² - 3a - 15 = 0
To solve this, we had to use the quadratic formula.
a = 3 ± √(9 + -4(1)(-15))
                    2(1)
a = 3 ± √(69)
            2
Now lets change the numbers a bit.2. a² - 2a - 15 = 0
(a - 5)(a = 3) = 0
a = 5, a = -3
^^ Here, instead of using the quadratic formula (which would've worked as well) we simply factored it. Like Mr. K says, why use the sledge hammer when we can use the hammer?

Lets stop for a moment and see if we recognize any patterns in the following equation.
3. (x - 2/x)² - 2(x - 2/x) - 15 = 0
This is something we call a quadratic in form.
If we're to do something like this:
Let a = x - 2/x
we end up with an equation that looks like this:
a² - 2a - 15 = 0
Which happens to be the exact same equation as the one we did above. So have you seen the pattern??

Above, we figured out that the roots of the equation a² - 2a - 15 = 0 were a = 5 and a = -3.
AND
We Let a = x - 2/x as well.
So lets plug in what we need to and solve for question 3.

a = 5
x - 2/x = 5
Now multiply both sides by x to get rid of the x in 2/x.
x² - 2 = 5x
x² - 5x - 2 = 0
^^ Since that doesn't factor nicely, this is where we can use the quadratic formula.
x = -b ± √(b² - 4ac
                2a
x = 5 ± √(25 - 4(1)(-2))
                   2
x = 5 ± √(23)
            2
x₁ = 5 + √(23)
             2
x₂ = 5 - √(23)
             2

a = -3
For this one, we do it exactly the same way.
If you guys want to try it out and want to see if you did it right, here's the answer.
x = 3 ± √(17)
             2

**SOMETHING TO GET INTO THE HABIT OF DOINGLets look at another quadtraic in form:
4. x⁴ - 13² + 36 = 0
- Let a = x²
- a² - 13a + 36 = 0
- (a - 9)(a = 4)
- a = 9, a = -4
It should be pretty straight forward on how to get there, but if you have any questions, feel free to ask! =)
Anyway, this is where most people do this:
x² = 9, x² = 4
x = 3, -3, x = 2, -2
9 OUT OF 10 TIMES when you solve it this way, you'll forget the negative (-) solution
SOOOO..
Lets get into the habit of solving our equation this way:
x² - 9 = 0, x² - 4 = 0
(x + 3)(x - 3) = 0, (x + 2)(x - 2) = 0
x = -3 and 3 , x = -2 and 2
This way, there's no way you'll forget the negative (-) solution.

Okay. I hope this helped Sam! =) Get well soon! .. If you have any questions don't be shy to ask. :D

NEXT SCRIBE IS .. How'd you guess? MERIAN.

Homework tonight is Exercise 15. Have fun =)