Solve x² - 3x - 2 = 0

To complete the square for the equation ax² + bx + c = 0 there are three steps.. 1. In order to complete the square, we need to take the first two parts of the quadtratic equation, and put them in the form of (    )²... Clearly we need an x in there to form the x² part of the equation... This creates (x + ?)².... The other part of the squared bracket is the coefficient (the b part). We need to half that then add it in... This creates (x + b/2)²... When the term in the brackets is squared, b/2 will form bx. Now we are left with the first part of our equation, (x + b/2)² 2. One problem... (x + b/2)² expanded makes x² + bx + (b/2)².. we don't want this (b/2)² term, its just rubbish created from factorising, so we must ensure we subtract it from our final equation.. In the final equation put -(b/2)² 3. Finally, we can write the equation in the form of completing the square as: (x + b/2)² + c - (b/2)² = 0, and solve the equation Now, lets apply this to our question.. x² - 3x - 2 = 0... First, put x² - 3x in the form of ( x + b/2 )² ... this makes ( x - 3/2 )² Then, minus - (b/2)².... this makes - ( - 3/2 )² Now, put it into the form of ( x + b/2 )² - ( b/2 )² + c = 0... this makes ( x - 3/2 )² - ( - 3/2)² - 2 = 0 Simplified is ( x - 3/2 )² - 9/4 - 2 = 0 Even more simplified is ( x - 3/2 )² - 17/4 = 0 So, lets find x! To complete the square for the equation ax² + bx + c = 0 there are three steps.. 1. In order to complete the square, we need to take the first two parts of the quadratic equation, and put them in the form of ( )²... Clearly, we need an x in there to form the x² part of the equation... This creates (x + ?)².... The other part of the squared bracket is the coefficient (the b part). We need to half that then add it in... This creates (x + b/2)²... When the term in the brackets is squared, b/2 will form bx. Now we are left with the first part of our equation, (x + b/2)² 2. One problem... (x + b/2)² expanded makes x² + bx + (b/2)².. we don't want this (b/2)² term, its just rubbish created from factorising, so we must ensure we subtract it from our final equation.. In the final equation put -(b/2)² 3. Finally, we can write the equation in the form of completing the square as: (x + b/2)² + c - (b/2)² = 0, and solve the equation Now, lets apply this to our question.. x² - 3x - 2 = 0... First, put x² - 3x in the form of ( x + b/2 )² ... this makes ( x - 3/2 )² Then, minus - (b/2)².... this makes - ( - 3/2 )² Now, put it into the form of ( x + b/2 )² - ( b/2 )² + c = 0... this makes ( x - 3/2 )² - ( - 3/2)² - 2 = 0 Simplified is ( x - 3/2 )² - 9/4 - 2 = 0 Even more simplified is ( x - 3/2 )² - 17/4 = 0 So, lets find x! Rearrange so that x is on one side and all the numbers are on the other.. ( x - 3/2 )² = 17/4 Then, square root both sides to free x from the brackets.. x - 3/2 = ∓ sqrt17 / sqrt4 Simplify further x – 3/2 = ∓ sqrt 17/2 Finally, make x the subject by moving 3/2 over.. x = (3+ sqrt17) / 2   or  x = (3 - sqrt17) / 2