 The Chinese also use the root procedure to solve non-monic quadratics and quadratics with negative coefficients. For example, we can try to solve 3x squared plus 150x equals 22353. So we'll set down the number, sides, and squares, and shift once, twice. And so we see our solution has three digits. We guess the first digit. Remember, it will be multiplied by the squares and added to the sides. And since we'll have to subtract it, that means the guess times the sum must be less than 22. So we guess 2, and our first pass, our guess 2 times the squares, 3, and add to the sides. Then our guess 2 times the sides, 750, and subtract from the number. And we'll do our second pass, our guess 2 times the squares, and add to the sides, and shift. The sides shift one place, and the squares shift two. And we'll guess the next digit of the root. We guess 5, and our first pass, guess times squares, and add to sides. Guess times sides, and subtract from number, that's too big. And so this means we have to reset and guess lower. And so we guess 4, and our first pass, guess times squares, and add to sides. Then guess times sides, and subtract from number. Then guess times squares, and add to sides, and shift. And finally we'll guess our third digit, 9, and take our first pass. Guess times squares, and add to sides. Guess times sides, and subtract from number. And at this point, since there is nothing left of the number, we have our solution 249. Or if we have a negative coefficient, so remember the Chinese distinguish between positive and negative numbers by whether they used red or black counting rods. We'll just use our standard negative sign, set down the coefficients, and shift. Since our sides coefficient is negative, then if it remains negative after we add the guess times the square, it will only increase the number. So our guess has to be large enough to make this coefficient positive. And so we guess 8, and our first pass, guess times squares, and add to sides. Then guess times sides, and subtract from number. For our second pass, guess times squares, and add to sides, and shift. And notice that our coefficient is now positive. So at this point, everything proceeds as before, we guess the next digit, make our first pass, guess times squares, and add to sides. Then guess times sides, and subtract from number. Our second pass, guess times squares, and add to sides, and shift. Now to avoid running out of room, we'll shift everything over a couple of places. And then we'll move the sides one place and the squares two places. And we guess the next digit, 1, we make our first pass, we make our second pass, and shift. Again, we'll move everything over so we have enough room to shift, and then move the sides one place and the squares two places. And we guess the next digit, 2, and do our first pass, guess times squares, and add to sides. Guess times sides, and subtract from number. And while we have found the solution to two decimal places, 83.12, we should set up for the next place anyway. So we'll engage in our second pass.