 In 1769, Lagrange published On the Solution of Numerical Equations. The paper discussed the theory of equations and presented a numerical method based on continued fractions. To find a real solution to a polynomial equation, Lagrange proceeded as follows. First, find, if possible, two consecutive integers p and p plus 1, so a root x is greater than p and less than p plus 1. A root x is p plus 1 over y, where y is a new variable, and substitute this into the equation, yielding a new equation. By construction, 1 over y is greater than 0 and 1 over y is less than 1, so y is greater than 0, actually, y is greater than 1. So find consecutive integers q and q plus 1, so a root y is greater than q, and y is less than q plus 1, and Laver rins repeat. So let T as Lagrange's method to find the first three convergence of a solution to x squared plus 5x minus 18 equals 0. So we can evaluate our polynomial at 1, 2, 3, and so on, which gives us, since the expression is negative at x equals 2 and positive at x equals 3, there must be a root between 2 and 3. So we'll let x equal 2 plus 1 over y, and we substitute that into our equation we get. So again, evaluating our new equation at 1, 2, 3, and so on, we find, so there is a root between y equals 2 and 3. So let y equal 2 plus 1 over z, and substitute, and again, we'll evaluate and find there is a root between z equals 2 and z equals 3. So we'll let z equal 2 plus 1 over w, but let's put our results together. x is 2 plus 1 over y, but y is 2 plus 1 over z, but z is 2 plus 1 over w, and so on. And so we have our continued fraction expansion for our solution x, and we can find our convergence. The first convergent is just the whole number part. The next convergent will incorporate the first fraction, one half, and get. We'll incorporate a little bit more of the continued fraction and get. Now remember, the convergence are ultimately larger and smaller than the limit, so we conclude that our solution is someplace between 5 halves and 12 fifths. And in fact, if we compute our polynomial at these two points we find, so the root is between these two values.