 We can complete the square on the polynomial equation x squared plus bx equals c by adding b over 2 squared to both sides. But what about ax squared plus bx equals c? And the useful thing to remember is that you can always use algebra to rewrite an equation. We know how to solve this equation, so let's rewrite this equation so it looks like the equation we can solve. So, for example, let's solve by completing the square 3x squared plus 6ax equals 22. So the problem is we know what to do if we have a monic polynomial if our coefficient of x squared is just 1. Well, notice that if we divide all terms by 3, we get a monic polynomial. So let's divide all terms by 3, and now we can complete the square by adding the square of half the coefficient of x. So our coefficient of x is 2. The square of half will be 1, and so we'll add 1 to both sides. Now remember, the reason we did all of this work is so that our left-hand side is a perfect square. So the left-hand side is going to be the square of something. The right-hand side we'll have to add, and now we have square equal to number, so we can take the square root of both sides. Don't forget we have to use plus or minus, and we can solve for x by subtracting 1 from both sides. Or again, let's take an equation like this. First, we have to isolate the variable terms. If only this coefficient of x squared were 1 instead of 2. Well, the nice thing about mathematics is we don't have to just engage in wishful thinking. We can do something about it, so we'll divide by the coefficient of x squared to get a monic polynomial. So we'll divide everything by 2, then we'll add the square of half the coefficient of x. So we're going to add the square of 5 halves to both sides. We went through all of this work so that our left-hand side was a perfect square. So we'll rewrite it as a perfect square. Our right-hand side is the sum of some fractions, so we'll go ahead and compute that as well. Take the square root of both sides, and solve for x.