Ejercicio 279

octubre 30, 2019abril 15, 2019 por Gestor Baldor

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CAPITULO XXXV

Dadas la suma y el producto de dos números, hallar los números

Ejercicio 279

Encontrar dos números sabiendo que:

La suma es 11 y el producto 30
$\begin{matrix} x^{2} - 11 x + 30 & = 0 \\ (x - 6) (x - 5) & = 0 \\ x - 6 & = 0 \\ x_{1} & = 6 \\ x - 5 & = 0 \\ x_{2} & = 5 \end{matrix}$
La suma es -33 y el producto 260
$\begin{matrix} x^{2} + 33 x + 260 & = 0 \\ (x + 13) (x + 20) & = 0 \\ x + 13 & = 0 \\ x_{1} & = - 13 \\ x + 20 & = 0 \\ x_{2} & = - 20 \end{matrix}$
La suma es -1 y el producto -306
$\begin{matrix} x^{2} + x - 306 & = 0 \\ (x + 18) (x - 17) & = 0 \\ x + 18 & = 0 \\ x_{1} & = - 18 \\ x - 17 & = 0 \\ x_{2} & = 17 \end{matrix}$
La suma es -49 y el producto 294
$\begin{matrix} x^{2} + 49 x + 294 & = 0 \\ (x + 42) (x + 7) & = 0 \\ x + 42 & = 0 \\ x_{1} & = - 42 \\ x + 7 & = 0 \\ x_{2} & = - 7 \end{matrix}$
La suma es 6 y el producto -247
$\begin{matrix} x^{2} - 6 x - 247 & = 0 \\ (x - 19) (x + 13) & = 0 \\ x - 19 & = 0 \\ x_{1} & = 19 \\ x + 13 & = 0 \\ x_{2} & = - 13 \end{matrix}$
La suma es $\frac{3}{2}$ y el producto -1
$\begin{matrix} x^{2} - \frac{3}{2} x - 1 & = 0 \\ 2 x^{2} - 3 x - 2 & = 0 \\ 2 x^{2} - 4 x + x - 2 & = 0 \\ 2 x (x - 2) + (x - 2) & = 0 \\ (x - 2) (2 x + 1) & = 0 \\ x - 2 & = 0 \\ x_{1} & = 2 \\ 2 x + 1 & = 0 \\ 2 x & = - 1 \\ x_{2} & = - \frac{1}{2} \end{matrix}$
La suma es $- \frac{22}{3}$ y el producto 8
$\begin{matrix} x^{2} + \frac{22}{3} x + 8 & = 0 \\ 3 x^{2} + 22 x + 24 & = 0 \\ 3 x^{2} + 18 x + 4 x + 24 & = 0 \\ 3 x (x + 6) + 4 (x + 6) & = 0 \\ (x + 6) (3 x + 4) & = 0 \\ x + 6 & = 0 \\ x_{1} & = - 6 \\ 3 x + 4 & = 0 \\ 3 x & = - 4 \\ x_{2} & = - \frac{4}{3} \end{matrix}$
La suma es $\frac{1}{4}$ y el producto $- \frac{3}{8}$
$\begin{matrix} x^{2} - \frac{1}{4} x - \frac{3}{8} & = 0 \\ 8 x^{2} - 2 x - 3 & = 0 \\ 8 x^{2} + 4 x - 6 x - 3 & = 0 \\ 4 x (2 x + 1) - 3 (2 x + 1) & = 0 \\ (2 x + 1) (4 x - 3) & = 0 \\ 2 x + 1 & = 0 \\ 2 x & = - 1 \\ x_{1} & = - \frac{1}{2} \\ 4 x - 3 & = 0 \\ 4 x & = 3 \\ x_{2} & = \frac{3}{4} \end{matrix}$
La suma es $- 13 \frac{4}{7}$ y el producto -6
$\begin{matrix} - 13 \frac{4}{7} & \leftrightarrow - \frac{95}{7} \\ x^{2} + \frac{95}{7} x - 6 & = 0 \\ 7 x^{2} + 95 x - 42 & = 0 \\ 7 x^{2} + 98 x - 3 x - 42 & = 0 \\ 7 x (x + 14) - 3 (x + 14) & = 0 \\ (x + 14) (7 x - 3) & = 0 \\ x + 14 & = 0 \\ x_{1} & = - 14 \\ 7 x - 3 & = 0 \\ 7 x & = 3 \\ x_{2} & = \frac{3}{7} \end{matrix}$
La suma es $- 3 \frac{1}{3}$ y el producto 1
$\begin{matrix} - 3 \frac{1}{3} & \leftrightarrow - \frac{10}{3} \\ x^{2} + \frac{10}{3} x + 1 & = 0 \\ 3 x^{2} + 10 x + 3 & = 0 \\ 3 x^{2} + 9 x + x + 3 & = 0 \\ 3 x (x + 3) + (x + 3) & = 0 \\ (x + 3) (3 x + 1) & = 0 \\ x + 3 & = 0 \\ x_{1} & = - 3 \\ 3 x + 1 & = 0 \\ 3 x & = - 1 \\ x_{2} & = - \frac{1}{3} \end{matrix}$
La suma es $\frac{31}{40}$ y el producto $\frac{3}{20}$
$\begin{matrix} x^{2} - \frac{31}{40} x + \frac{3}{20} & = 0 \\ 40 x^{2} - 31 x + 6 & = 0 \\ 40 x^{2} - 15 x - 16 x + 6 & = 0 \\ 5 x (8 x - 3) - 2 (8 x - 3) & = 0 \\ (8 x - 3) (5 x - 2) & = 0 \\ 8 x - 3 & = 0 \\ 8 x & = 3 \\ x_{1} & = \frac{3}{8} \\ 5 x - 2 & = 0 \\ 5 x & = 2 \\ x_{2} & = \frac{2}{5} \end{matrix}$
La suma es $- \frac{1}{6}$ y el producto $- \frac{5}{9}$
$\begin{matrix} x^{2} + \frac{1}{6} x - \frac{5}{9} & = 0 \\ 18 x^{2} + 3 x - 10 & = 0 \\ 18 x^{2} + 15 x - 12 x - 10 & = 0 \\ 3 x (6 x + 5) - 2 (6 x + 5) & = 0 \\ (6 x + 5) (3 x - 2) & = 0 \\ 6 x + 5 & = 0 \\ 6 x & = - 5 \\ x_{1} & = - \frac{5}{6} \\ 3 x - 2 & = 0 \\ 3 x & = 2 \\ x_{2} & = \frac{2}{3} \end{matrix}$
La suma es $\frac{7}{20}$ y el producto $- \frac{3}{10}$
$\begin{matrix} x^{2} - \frac{7}{20} x - \frac{3}{10} & = 0 \\ 20 x^{2} - 7 x - 6 & = 0 \\ 20 x^{2} - 15 x + 8 x - 6 & = 0 \\ 5 x (4 x - 3) + 2 (4 x - 3) & = 0 \\ (4 x - 3) (5 x + 2) & = 0 \\ 4 x - 3 & = 0 \\ 4 x & = 3 \\ x_{1} & = \frac{3}{4} \\ 5 x + 2 & = 0 \\ 5 x & = - 2 \\ x_{2} & = - \frac{2}{5} \end{matrix}$
La suma es $4 \frac{1}{5}$ y el producto -4
$\begin{matrix} 4 \frac{1}{5} & \leftrightarrow \frac{21}{5} \\ x^{2} - \frac{21}{5} x - 4 & = 0 \\ 5 x^{2} - 21 x - 20 & = 0 \\ 5 x^{2} - 25 x + 4 x - 20 & = 0 \\ 5 x (x - 5) + 4 (x - 5) & = 0 \\ (x - 5) (5 x + 4) & = 0 \\ x - 5 & = 0 \\ x_{1} & = 5 \\ 5 x + 4 & = 0 \\ 5 x & = - 4 \\ x_{2} & = - \frac{4}{5} \end{matrix}$
La suma es $\frac{59}{72}$ y el producto $\frac{1}{6}$
$\begin{matrix} x^{2} - \frac{59}{72} x + \frac{1}{6} & = 0 \\ 72 x^{2} - 59 x + 12 & = 0 \\ 72 x^{2} - 27 x - 32 x + 12 & = 0 \\ 9 x (8 x - 3) - 4 (8 x - 3) & = 0 \\ (8 x - 3) (9 x - 4) & = 0 \\ 8 x - 3 & = 0 \\ 8 x & = 3 \\ x_{1} & = \frac{3}{8} \\ 9 x - 4 & = 0 \\ 9 x & = 4 \\ x_{2} & = \frac{4}{9} \end{matrix}$
La suma es $2$ y el producto -4
$\begin{matrix} x^{2} - 2 x - 4 & = 0 \\ x & = \frac{- (- 2) \pm \sqrt{{(- 2)}^{2} - 4 (1) (- 4)}}{2 (1)} \\ x & = \frac{2 \pm \sqrt{4 + 16}}{2} \\ x & = \frac{2 \pm \sqrt{20}}{2} \\ x & = \frac{2 \pm \sqrt{2^{2} .5}}{2} \\ x & = \frac{2 \pm 2 \sqrt{5}}{2} \\ x & = \frac{2 (1 \pm \sqrt{5})}{2} \\ x_{1} & = 1 + \sqrt{5} \\ x_{2} & = 1 - \sqrt{5} \end{matrix}$
La suma es 1 y el producto $- \frac{11}{4}$
$\begin{matrix} x^{2} - x - \frac{11}{4} & = 0 \\ 4 x^{2} - 4 x - 11 & = 0 \\ x & = \frac{- (- 4) \pm \sqrt{{(- 4)}^{2} - 4 (4) (- 11)}}{2 (4)} \\ x & = \frac{4 \pm \sqrt{16 + 176}}{8} \\ x & = \frac{4 \pm \sqrt{192}}{8} \\ x & = \frac{4 \pm \sqrt{8^{2} .3}}{8} \\ x & = \frac{4 \pm 8 \sqrt{3}}{8} \\ x & = \frac{4 (1 \pm 2 \sqrt{3})}{} \\ x & = \frac{1 \pm 2 \sqrt{3}}{2} \\ x_{1} & = \frac{1 + 2 \sqrt{3}}{2} \\ x_{2} & = \frac{1 - 2 \sqrt{3}}{2} \end{matrix}$
La suma es $- 1 \frac{1}{3}$ y el producto $- 6 \frac{5}{9}$
$\begin{matrix} - 1 \frac{1}{3} \leftrightarrow - \frac{4}{3} \\ - 6 \frac{5}{9} \leftrightarrow - \frac{59}{9} \\ x^{2} + \frac{4}{3} x - \frac{59}{9} & = 0 \\ 9 x^{2} + 12 x - 59 & = 0 \\ x & = \frac{- 12 \pm \sqrt{1 2^{2} - 4 (9) (- 59)}}{2 (9)} \\ x & = \frac{- 12 \pm \sqrt{144 + 2124}}{18} \\ x & = \frac{- 12 \pm \sqrt{2268}}{18} \\ x & = \frac{- 12 \pm \sqrt{2^{2} . 3^{4} .7}}{18} \\ x & = \frac{- 12 \pm 2 (3^{2}) \sqrt{7}}{18} \\ x & = \frac{- 12 \pm 18 \sqrt{7}}{18} \\ x & = \frac{6 (- 2 \pm 3 \sqrt{7})}{} \\ x & = \frac{- 2 \pm 3 \sqrt{7}}{3} \\ x_{1} & = \frac{- 2 + 3 \sqrt{7}}{3} \\ x_{2} & = \frac{- 2 - 3 \sqrt{7}}{3} \end{matrix}$
La suma es $a$ y el producto $- 2 a^{2}$
$\begin{matrix} x^{2} - a x - 2 a^{2} & = 0 \\ (x - 2 a) (x + a) & = 0 \\ x - 2 a & = 0 \\ x_{1} & = 2 a \\ x + a & = 0 \\ x_{2} & = - a \end{matrix}$
La suma es $- 7 b$ y el producto $10 b^{2}$
$\begin{matrix} x^{2} + 7 b x + 10 b^{2} & = 0 \\ (x + 5 b) (x + 2 b) & = 0 \\ x + 5 b & = 0 \\ x_{1} & = - 5 b \\ x + 2 b & = 0 \\ x_{2} & = - 2 b \end{matrix}$
La suma es $\frac{m}{2}$ y el producto $- \frac{m^{2}}{9}$
$\begin{matrix} x^{2} - \frac{m}{2} x - \frac{m^{2}}{9} & = 0 \\ 18 x^{2} - 9 m x - 2 m^{2} & = 0 \\ 18 x^{2} + 3 m x - 12 m x - 2 m^{2} & = 0 \\ 3 x (6 x + m) - 2 m (6 x + m) & = 0 \\ (6 x + m) (3 x - 2 m) & = 0 \\ 6 x + m & = 0 \\ 6 x & = - m \\ x_{1} & = - \frac{m}{6} \\ 3 x - 2 m & = 0 \\ 3 x & = 2 m \\ x_{2} & = \frac{2 m}{3} \end{matrix}$