| So now we have a great way to solve equations that look like: | ||
|
2X2 + 4X - 6 = 0 |
||
| What about stuff like ... | ||
|
|
||
| If you look really close, | ||
| you will see that the exponents on the X terms on the left | ||
| are exactly the squares of the exponents on the X terms in the center. | ||
| We know that ... | ||
|
X2 = X × X |
||
| But these are also squares ... | ||
|
|
||
| Big deal. So what? | ||
| OK, here's what we do. | ||
| We chose a letter (usually "u") | ||
| to stand for the X term in the middle of the equation. | ||
| That way, the X term on the left is u2. | ||
| Here goes ... | ||
| Example: | ||
| Solve this puppy ... | ||
|
|
||
| Now substitute "u" and "u 2" into the equation for the X's ... | ||
|
u2 - 6u + 5 = 0 |
||
| The Quadratic Formula works for any letter, not just X. | ||
| That means we can use it here. | ||
| Lucky us! | ||
|
|
||
|
|
||
But u =  ,
so now put the back
in for u and find X ... |
||
|
|
||
| Let's do another one ... | ||
| Example: | ||
|
|
||
| u = X 2 so we put X 2 back in to find X ... | ||
|
|
||
| Since there are not any real numbers that you can square | ||
| to get a negative three, that one has no answer. | ||
| So the only answer to this one is X = 1 | ||
| These things could get really involved. | ||
| The trick is to just look at the thing you replace with the u | ||
| as a glob. | ||
| Then just forget about it until you get done | ||
| with the rest of the problem. | ||
| Here's a nasty one. | ||
| Example: | ||
|
4(2X2 + 6X)2 + 12(2X2 + 6X) + 5 = 0 |
||
| Substitute ... | ||
|
u = (2X2 + 6) so u2 = (2X2 + 6)2 |
||
| We have ... | ||
|
|
||
| So great, we found u. | ||
| Now we need to put back the thing that we substituted for | ||
| (2X 2 + 6X) so ... | ||
|
|
||
| Now move the numbers over to the left side of the equation | ||
| so that we can use the Quadratic Formula ... | ||
|
|
||
| So ... | ||
|
|
||
| That one was pretty messy. | ||
| But the deal is that no matter what kind of mess you have, | ||
| if the variable glob on the left is the square | ||
| of the variable glob in the middle you can use this trick. | ||
|
copyright 2005 Bruce Kirkpatrick |
