Completing the Square

Why? Why? Why are we learning it? Why? Why? Why? Completing the Square allows us to solve unfactorable quadratic equations.

There will be two types of problems: x2 + bx + c = 0 ax2 + bx + c = 0

Solving x2 + bx + c = 0 Process: Move the constant. Determine what must be added to form a “perfect square”. Factor. Square root. Solve what is left.

x2 + 6x + 2 = 0 -2 -2 x2 + 6x + = -2 Move the constant. Putting it into action! x2 + 6x + 2 = 0 -2 -2 x2 + 6x + = -2 Move the constant. Subtract 2 from each side. Leave a space where the 2 was, we’ll fill it in shortly.

x2 + 6x + = -2 (3)2 + 9 Determine what must be added to form a “perfect square”. How to do it: Take half of “b”. 6/2 = 3 Square this number. (3)2 = 9 Add to both sides.

x2 + 6x + (3)2 = -2 + 9 ( )2 = 7 x + 3 Factor. ( )2 = 7 x + 3 Factor. The left side is now a perfect square trinomial.

(x + 3)2 = 7 x + 3 = 7  Square root!! Remember the “”

 x = -3 7  x + 3 = 7 -3 -3 Solve what is left. -3 -3 x = -3 7  Solve what is left. Subtract 3 from both sides.

One more time: x2 + 6x + 2 = 0 x2 + 6x = -2 x2 + 6x + (3)2 = -2 + 9 Move the constant. x2 + 6x = -2 Form a “perfect square”. x2 + 6x + (3)2 = -2 + 9 Factor. (x + 3)2 = 7 (x + 3)2 = 7 Square root. x + 3 = ± 7 x = -3 ± 7 Solve what is left.

Solving ax2 + bx + c = 0 Process: Divide each term by “a”. Move the constant. Form a “perfect square”. Factor. Square root. Solve what is left.

2x2 + 8x - 3 = 0 2 2 2 2 x2 + 4x - = 0 3 2 Divide each term by “a”. Doing it! 2x2 + 8x - 3 = 0 2 2 2 2 x2 + 4x - = 0 3 2 Divide each term by “a”. Divide each term by 2. Don’t use decimals.

x2 + 4x - = 0 + + x2 + 4x + = 3 2 3 2 3 2 Move the constant. + + x2 + 4x + = 3 2 Move the constant. Add 3/2 to each side. Leave a space where the -3/2 was, we’ll fill it in shortly.

x2 + 4x + = 3 2 (2)2 + 4 Determine what must be added to form a “perfect square”. How to do it: Take half of “b”. 4/2 = 2 Square this number. (2)2 = 4 Add to both sides.

x2 + 4x + (2)2= + 4 3 2 ( )2 = 11 2 x + 2 Factor. The left side is now a perfect square trinomial.

(x + 2)2 = 11 2 x + 2 =  11 2 Square root!! Remember the “”

x = -2  x + 2 =  -2 -2 x = -2  11 2 11 2 22 2 Solve what is left. -2 -2 x = -2 11 2  x = -2  22 2 Solve what is left. Subtract 2 from both sides. Rationalize the denominator.

Solving ax2 + bx + c = 0 Process: Divide each term by “a”. Move the constant. Form a “perfect square”. Factor. Square root. Solve what is left.

 The End!!!                