| So now we have a step by step process to find X, | ||
| but wow! it is really long and involved. | ||
| It is a lot to remember! | ||
| Couldn't we maybe just have a formula | ||
| to solve these things all in one shot? | ||
| Huh? | ||
| Please? | ||
| SURE | ||
| For the last two pages, we've been solving stuff like: | ||
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3X2 + 5X + 2 = 0 |
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| and | ||
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X2 - 4X + 3 = 0 |
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| Here's what we're going to do. | ||
| We're going to write one of these equations as ... | ||
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AX2 + BX + C = 0 |
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| Then we're going to do our long process from the last page on it. | ||
| In the end, we will have ... | ||
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X = some junk with A's B's and C's in it |
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| From then on, when we have one of these equations | ||
| all we have to do is pick out the A, B, and C value | ||
| and put it in the formula we are about to build. | ||
| No more nasty factoring. | ||
|
Just remember the formula, and you're set. |
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| Here we go: | ||
| STEP 1: | ||
| Divide both sides of the equation by the coefficient on the X2 ... | ||
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| Simplify ... | ||
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| STEP 2: | ||
| Take the X term and determine the value of B. | ||
| Use that to find B2. | ||
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| STEP 3: | ||
| Put the B 2 value into the equation. | ||
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| STEP 4: | ||
| Gather up the terms of the square, | ||
| and write them as the square. | ||
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| STEP 5: | ||
| Put all of the other baloney that's not part of the square | ||
| on the right side of the equation. | ||
| Simplify it as much as you can. | ||
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| STEP 6: | ||
| Solve this for X. | ||
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|
||
| And there it is! | ||
| The most important formula in all of Algebra. | ||
| It was a lot of work getting here, | ||
| but here is the payoff. | ||
| Now, if we have something like ... | ||
|
5X2 - 13X + 6 = 0 |
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| We can say, OK: | ||
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A = 5 B = -13 C = 6 |
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| and put these numbers into "THE FORMULA" | ||
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| This is the same answer as we got before for this one, | ||
| but this time it was a whole lot easier. | ||
| Our new formula is called ... | ||
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The Quadratic Formula |
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| When we use it, sometimes we will get two answers, | ||
| sometimes we will get one answer, | ||
| and sometimes we will not get any answers. | ||
| If we don't get any answers, | ||
| it doesn't mean that the formula didn't work. | ||
| It means that the problem really doesn't have any answers. | ||
| There is a quick way to tell how many answers you will get. | ||
| Just look at this part of the formula: | ||
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| If B 2 - 4AC works out to be a positive number, | ||
| you will get two answers for X. | ||
| If B 2 - 4AC works out to be zero, | ||
| you will get one answer for X. | ||
| If B 2 - 4AC works out to be a negative number, | ||
| you will not get any answers for X. | ||
| Because this piece of the formula can tell you that, | ||
| it gets it's own name. | ||
| It is called: | ||
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The Discriminant |
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| Let's do another one ... | ||
| Example: | ||
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4X2 - 20X + 25 = 0 |
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| So: | ||
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A = 4 B = -20 C = 25 |
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| The "discriminant" part of the formula was equal to zero, | ||
| so we only get one answer. | ||
| Example: | ||
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6X2 = - 2X + 4 |
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| Whoa! Time out! | ||
| Before we do anything, we need to move all of the values | ||
| over to the left side of the equation so we have: | ||
|
AX2 + BX + C = 0 |
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|
|
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| So: | ||
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A = 6 B = 2 C = -4 |
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|
copyright 2005 Bruce Kirkpatrick |
