Is it really a discovery?

33

After considering approximate numerical solutions of the equation of calculated powers using the methods of trials, chords, tangents and integration, in the final elective lesson on the topic: “Approximate solutions of the equation,” students tried to explore the following method for numerically solving the equation.

Let the equation $f(x) = 0$ have a single root $x_0$ on the segment $a \leq x \leq b$, and let the function $f(x)$ be continuous on this segment and take values ​​of different signs at its ends. Let us draw chords $AC$ and $BC$ for the graph of the given function on the segment $[a, b]$.

Let us write down the equation of the line passing through the points $A[b; f(b)]$ and $C(x_0; 0)$:

$$\frac{f(b) - y}{f(b) - 0} = \frac{b - x}{b - x_0},$$

from which it is located

$$y = f(b) - \frac{b - x}{b - x_0} f(b).$$

Similarly, we write the equation of the line passing through the points $B[a; f(a)]$ and $C(x_0; 0)$:

$$\frac{f(a) - y}{f(a) - 0} = \frac{a - x}{a - x_0},$$

Graph of the function $y = f(b) - \frac{b - x}{b - x_0} f(b)$

where

$$y = f(a) - \frac{a - x}{a - x_0} f(a).$$

To find the coordinates of the intersection point of these lines, we solve the system

$$\begin{cases} y = f(b) - \frac{b - x}{b - x_0} f(b), \\ y = f(a) - \frac{a - x}{a - x_0} f(a). \end{cases}$$

Writing out the whole thing 收支

$$f(b) - \frac{b - x}{b - x_0} f(b) = f(a) - \frac{a - x}{a - x_0} f(a),$$

$$f(b) \left[ 1 - \frac{b - x}{b - x_0} \right] = f(a) \left[ 1 - \frac{a - x}{a - x_0} \right],$$

$$f(b) \cdot \frac{x - x_0}{b - x_0} = f(a) \cdot \frac{x - x_0}{a - x_0},$$

$$\frac{f(b)}{f(a)} = \frac{b - x_0}{a - x_0},$$

we find

$$x_0 = \frac{bf(a) - af(b)}{f(a) - f(b)}.$$

A formula has been obtained for the root of this equation $f(x) = 0$! Learning from the unexpected result with great excitement and enthusiasm, they even exclaimed: “Eureka!”

But after a few minutes, an alarming silence was restored in the audience. Everyone was stunned and everyone asked: “How can this be?” Prominent scientists - mathematicians Abel and Evariste Galya for a comprehensive proof that for all equations above the fourth degree it is impossible to create formulas for exact solutions.

Where was the mistake made?

Y. M. Kleiman