| Once upon a time a few pages ago, | ||
| I promised you a method to factor some equations directly | ||
| without all of the guessing games. | ||
| Here it is. | ||
| Say you have an equation that looks something like ... | ||
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6X2 + 2X - 4 = 0 |
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| We will be trying to turn this into ... | ||
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(X ± some number)2 = some other number |
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| So how do we do it? | ||
| Well, when we had (X + 4)2 and multiplied it out, | ||
| we got ... | ||
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(X + 4)2 = X2 + 8X + 16 |
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| In general, using the letter "B" to stand for | ||
| any old number that we might have | ||
| we get ... | ||
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(X + B)2 = X2 + 2BX + B2 |
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| This is what we need to get to. | ||
| We take whatever we have at the start | ||
| and make it look like X 2 + 2BX + B 2 | ||
| So looking at 6X 2 + 2X - 4 = 0, | ||
| the first thing we need to do is get rid of the 6 part of the 6X 2. | ||
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To do that, we divide both sides by 6 ... |
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| STEP 1: | ||
| Divide both sides of the equation by the coefficient on the X 2 ... | ||
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| and simplify ... | ||
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| STEP 2: | ||
| Take the X term and determine the value of B. | ||
| Use that to find B 2. | ||
| In this problem, the X term is 1/3X. | ||
| Our formula says this is equal to 2BX. | ||
| Solve for B ... | ||
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| So if 1/6 is B then multiply it times itself | ||
| to get B 2. | ||
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| STEP 3: | ||
| Put the B 2 value into the equation. | ||
| We can do almost anything we want to an equation | ||
| if we do the same thing to both sides. | ||
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| STEP 4: | ||
| Gather up the terms of the square, | ||
| and write them as the square. | ||
| At this point, the first three terms on the left side | ||
| are the terms of our square. | ||
| That is, the (X + B) 2 = X 2 + 2BX + B 2 | ||
| We can substitute the square for these three terms ... | ||
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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. | ||
| We're finally at the place where we have ... | ||
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(X ± some number)2 = some other number |
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| From here it's easy. | ||
| We just peel the onion ... | ||
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| So we have two answers ... | ||
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| If you check back a few pages, | ||
| you'll see that we got the same answer there too! | ||
|
copyright 2005 Bruce Kirkpatrick |
