Quadratic Formula
Hello classmates, I will be scribing today's lesson (14/03/07) on solving equations using the quadratic formula. Let us begin this madness...First of all, we have to find the the quadratic formula! Oh what fun... so let us get started.
To begin our quest to find the quadratic formula, we must first know what the standard form of a quadratic is, which we all should know already, but for those who have forgotten here is what it is:
>>> ax² + bx + c = 0
Alrighty, now that we know what that is, we may now begin working our way to find that darn quadratic formula! So first, we must start by completing the square of that abomination, and if you have forgotten how to complete the square, then you did not pay attention during the complete the square lesson at all! So here is what should happen to the equation...
>>> x² + b/a x + c/a = 0
If you don't understand what I just did, all I did was divide everything by a.
>>> x² + b/a x = -c/a
It was necessary to move the "c/a" to the other side so that I can create the quadratic formula, mkay?
>>> x² + b/a x + b²/4a² = b²/4a² - c/a
I got the b²/4a² by dividing b/a by 2 and squaring it, you know? And to make sure that the equation is balanced, I added the b²/4a² to the other side of the equation too =D!
>>> (x + b/2a)² = b²/4a² - 4ac/4a²
The reason why the other side is all over 4a² now is because I took the common denominator to make it..err..easier..to..work with. Yeah...
Now that we've got that down, we now have to take the square root of both sides, you know the drill.
>>> x + b/2a = ± √ b² - 4ac I / 2a
>>> x = -b/2a ± √ b² - 4ac I / 2a
*drum rolls* Aaaaaaand..here's our quadratic formula!!
>>> x = -b ± √ b² - 4ac I / 2a
*Wipes sweat off face* Phew, now that we've got that out of the way, I will now show you something using our old friend Parabola!
Getting the zeroes using that formula is simple, as we all know. So..
x = -b + √ b² - 4ac I / 2a
= -b/2a + √ b² - 4ac I / 2a
x = -b - √ b² - 4ac I / 2a
= -b/2a - √ b² - 4ac I / 2a
That's all there is, and just one more interesting thing, the "-b/2a" is the value of the axis of symmetry and it is also the "x" value of the vertex. Cool eh?
Okay, I am not done yet! I'm almost..but not quite! I will now show you an example using the quadratic formula. There's going to be numbers this time, so no more of that a, b, c stuff..well..sorta.
Let's solve this equation:
>>> x² + 3x - 9 = 0
Since we've just started using the quadratic formula, it is a good habit to begin a question by first finding out what the a, b, c values are. If you're not following me, then to make it more understandable, look back at the quadratic formula and you will notice that there are a's, b's, and c's. So we need to find those values in order to substitute the values in so we can solve the zeroes or something. If you're not sure which number goes with which letter, then look back on the standard form of a quadratic.
>>> ax² + bx + c = 0
Looks like that equation we're trying to solve right? Meaning that..
>>> a = 1
>>> b = 3
>>> c = -9
Wahoo!! We got our values, now we can substitue them into the quadratic formula! Oh, and again, since we've just started using the formula, it is a good idea to write it down first before you start substituting things in, so that you can memorize it.
>>> x = -b ± √ b² - 4ac I / 2a
>>> x = -(3) ± √ (3)² - 4(1)(-9) I / 2(1)
>>> x = -3 ± √ 9 + 36 I / 2
>>> x = -3 ± √ 45 I / 2
Uh oh, there's no square root of 45! Well, we know what to do! We should all know that 9 is the biggest square root that can go into 45 and 9 x 5 = 45 so yarrr!
>>> x = -3 ± (√ 9 I) (√ 5 I) / 2
>>> x = -3 ± 3 √ 5 I / 2
Oh kay! Now we can find the roots >__>..
>>> x = -3 + 3 √ 5 I / 2
>>> x = -3 - 3 √ 5 I / 2
Yeah..and that's it! Pretty easy..M I RITE? Sure I am! Well now that that's over, remember these key things when using the quadratic formula.
1) Make sure to find your a, b, c values first. Makes life way easier.
2) Always write down the formula first! If you don't...you might forget it!
3) I don't know. ..This key thing is for you to decide! XD
Anyways, I'm done. I hate scribing and this took me over an hour to finish..rawr. Now that I'm finally done, it is time to choose the next scribe. *Looks at list and sees that mostly everyone has scribed already* Argh, I guess I'll choose....chrycel. Yeah..mkay.
Oh and btw, for those of you who could not attend class today, today's homework was Exercise 14, questions 1-6.
K then, I'm going to go relax now. Adieu!
posted by Kristina @ 7:52 PM
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