340. Find the positive root of the equation by construction:

$$
x^{2}+6 x=91
$$
340. Cardano's Solution: Let the square $F D$ (Fig. 74) be $x^{2}$, hence its side $F H=x$. $D G=D B=3$ (half the coefficient of $x$). Construct the square $A F E C$. The rectangle $A D$ equals the rectangle $D E$, i.e., equals $3 x$. The sum of the square $F D$ and the two rectangles equals $x^{2}+6 x$, which, by the condition, is 91. The small square $B C G D=9$, hence the square $A F E C=100$. Therefore, $A C=10$; but $A C=x+3$, so $x=7$. The usual method:

$x^{2}+6 x-91=0 ; x=-3 \pm \sqrt{9+91}=$

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-152.jpg?height=476&width=443&top_left_y=1561&top_left_x=1252)

Fig. 74.

$=-3 \pm 10 ; \quad x_{1}=7 ; \quad x_{2}=-13$