A way to solve when you can’t factor

To change the equation to have a perfect square What’s a perfect square? The first and last terms of the expression are squares, the middle term is double the product of the square roots. a 2 + 2ab + b 2 x 2 + 6x + 9 (x + 3) 2

When our x 2 term is a square and we can’t factor (remember, 1 is a square). 4x x + 7 = 0

Move the constant over to the other side. 4x x = -7

Figure out what the constant with the quadratic equation has to be. Technical terms incoming: The square of half the x-term divided by the square root of the x 2 term = ((2ab/2)/ √ (a 2 )) 2 In this problem, we have to add 9. ((12x/2) / √(4x 2 )) 2

A coefficient on our x 2 term makes things harder than they otherwise would be. If it was x 2 + 6x on the left side, what would we need the constant to be? 6/2 = 3, 3 2 = 9

4x x = -7 What were we adding to this again? Oh yeah, 9. ADD IT TO BOTH SIDES 4x x + 9 = = 2 Now we factor! (2x + 3) 2 = 2

Wait, this doesn’t equal 0. So what? Could we get rid of the square? YES Square root of both sides 2x + 3 = √(2)

Subtract 3 from both sides. 2x = √(2) – 3 Then divide both sides by 2 x = (√(2) – 3)/2 Wait, I thought we needed two answers? Square roots have a +/- implied