## How to solve quadratic equations by... COMPLETING THE SQUARE

 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 And that's it! The two answers to x2 + 5x - 9 = 0 are x = + 1.405 OR x = - 6.405

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 So if 3x2 - 11x - 8 = 0 then x = 4·288 OR x = - 0·622

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