COMPLETING THE SQUARE |
The QUADRATIC FORMULA (hidden behind a secret panel) |
Suppose you need to solve this grim looking quadratic equation (and we're warning you, the answers are not whole numbers!): | x2 + 5x - 9 = 0 |
First move the constant across. Here we do it by adding 9 to both sides: | x2 + 5x = 9     |
Now we're going to make a new equation by playing with the Left Hand Side (or LHS). We'll ignore the RHS for a moment. Follow these instructions exactly. | |
1/ divide the LHS by x . In this case we get | x + 5 |
2/ divide the number by 2 (don't divide the "x"). Now we get | x +2.5 |
3/ put both terms in a bracket, square it and then multiply it all out |
(x + 2.5)2
          = (x +2.5)(x +2.5)           = x2 + 2.5x + 2.5x + 6.25           = x2 + 5x + 6.25 |
Here comes the coolest part of the whole operation. Because we know that x2 + 5x =9 (look back a few lines , you'll find it written there!) we can swap the x2 +5x on the RHS for 9. | |
Now we've got our new equation | (x +2.5)2 = 9 + 6.25 |
Then a quick little sum gives us... | (x + 2·5)2 = 15·25 |
We now take the square root of both sides, and the clever bit is that square roots can be + or - | |
Here we get: |
              x + 2·5 = + sqrt(15.25)
OR         x + 2.5 = - sqrt(15.25) |
Grab a calculator to work out the square root... |
              x + 2·5 = + 3.905
OR         x + 2.5 = - 3.905 |
And then when you take away the 2·5 from both sides you get the two solutions |
x = +3·905 -2.5 = 1·405
OR x = -3·905 -2.5 = - 6·405 |
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Here's another one just to make sure you've got it. To make it more exciting this one has an x2 coefficient!
We're going to solve this little baby... | 3x2 - 11x - 8 = 0 |
Move the constant over | 3x2 - 11x = 8 |
Before we go on, we divide everything by the x2 coefficient because we want the x2 by itself | x2 - 11x/3 = 8/3     |
Here's where we start making our new equation... | |
Make the LHS into a square using steps 1,2 and 3 as before. (So divide through by x, then divide the constant by 2, then put the answer in a bracket and square it.) | (x - 11/6)2 |
Multiply the square out |
(x - 11/6)2 = x2 - 11x/6 - 11x/6 +121/36
                  = x2 - 11x/3 + 121/36 |
Now it's time to play the cool little trick! We know from before that x2 - 11x/3 = 8/3 so we put this into the RHS... | |
... and here's the new equation: | (x - 11/6)2 = 8/3 + 121/36 |
At this point we'll make everything into decimals: | (x - 1.833)2 = 2.667 + 3.361 = 6.028 |
Take square roots of both sides |
x - 1.833 = + sqrt(6.028)
OR x - 1.833 = - sqrt(6.028) |
Get the calculator and work out the square root... |
x - 1.833 = + 2.455
OR x - 1.833 = - 2.455 |
If we add 1.833 to both sides we get you get the two answers: |
x = +1·833 + 2·455 = 4·288
OR x = +1·833 -2.455 = - 0·622 |