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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. a**^{2} - 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. a**^{2} - __2__a - 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:

**a**^{2} - 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 **a**^{2} - 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**.

x^{2} - 2 = 5x

x^{2} - 5x - 2 = 0

^^ Since that doesn't factor nicely, this is where we can use the quadratic formula.

x = __-b ± √(b__^{2} - 4ac

2a

x = __5 ± √(25 - 4(1)(-2))__

2

x = __5 ± √(23)__

2

x_{1} = __5 + √(23)__

2

x_{2} = __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 DOING***Lets look at another quadtraic in form:*

**4. x**^{4} - 13^{2} + 36 = 0

**-** Let a = x^{2}

**-** a^{2} - 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:

x^{2} = 9, x^{2} = 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:

x^{2} - 9 = 0, x^{2} - 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 =)
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