Ejercicio 280 – Solucionario Baldor

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CAPITULO XXXV

Descomponer un trinomio en factores hallando sus raices

Ejercicio 280

Descomponer en factores, hallando las raíces:

Solución:

Se iguala la ecuación a 0
Se buscan las raíces
Se remmplazan los valores en la fórmula $a x^{2} + b x + c = a (x - x_{1}) (x - x_{2})$

$\begin{matrix} x^{2} - 16 x + 63 \\ x^{2} - 16 x + 63 & = 0 \\ (x - 9) (x - 7) & = 0 \\ x - 9 & = 0 \\ x_{1} & = 9 \\ x - 7 & = 0 \\ x_{2} & = 7 \\ x^{2} - 16 x + 63 & = (x - 9) (x - 7) \end{matrix}$
$\begin{matrix} x^{2} + 24 x + 143 \\ x^{2} + 24 x + 143 & = 0 \\ (x + 13) (x + 11) & = 0 \\ x + 13 & = 0 \\ x_{1} & = - 13 \\ x + 11 & = 0 \\ x_{2} & = - 11 \\ x^{2} + 24 x + 143 & = (x + 13) (x + 11) \end{matrix}$
$\begin{matrix} x^{2} - 26 x - 155 \\ x^{2} - 26 x - 155 & = 0 \\ (x - 31) (x + 5) & = 0 \\ x - 31 & = 0 \\ x_{1} & = 31 \\ x + 5 & = 0 \\ x_{2} & = - 5 \\ x^{2} - 26 x - 155 & = (x - 31) (x + 5) \end{matrix}$
$\begin{matrix} 2 x^{2} + x - 6 \\ 2 x^{2} + x - 6 & = 0 \\ 2 x^{2} + 4 x - 3 x - 6 & = 0 \\ 2 x (x + 2) - 3 (x + 2) & = 0 \\ (x + 2) (2 x - 3) & = 0 \\ x + 2 & = 0 \\ x_{1} & = - 2 \\ 2 x - 3 & = 0 \\ 2 x & = 3 \\ x_{2} & = \frac{3}{2} \\ 2 x^{2} + x - 6 & = 2 (x + 2) (x - \frac{3}{2}) \\ 2 x^{2} + x - 6 & = 2 (x + 2) (\frac{2 x - 3}{2}) \\ 2 x^{2} + x - 6 & = (x + 2) (2 x - 3) \end{matrix}$
$\begin{matrix} 12 x^{2} + 5 x - 2 \\ 12 x^{2} + 5 x - 2 & = 0 \\ 12 x^{2} - 3 x + 8 x - 2 & = 0 \\ 3 x (4 x - 1) + 2 (4 x - 1) & = 0 \\ (4 x - 1) (3 x + 2) & = 0 \\ 4 x - 1 & = 0 \\ 4 x & = 1 \\ x_{1} & = \frac{1}{4} \\ 3 x + 2 & = 0 \\ 3 x & = - 2 \\ x_{2} & = - \frac{2}{3} \\ 12 x^{2} + 5 x - 2 & = 12 (x - \frac{1}{4}) (x + \frac{2}{3}) \\ 12 x^{2} + 5 x - 2 & = 3 (\frac{4 x - 1}{4}) (\frac{3 x + 2}{3}) \\ 12 x^{2} + 5 x - 2 & = (4 x - 1) (3 x + 2) \end{matrix}$
$\begin{matrix} 5 x^{2} + 41 x + 8 \\ 5 x^{2} + 41 x + 8 & = 0 \\ 5 x^{2} + 40 x + x + 8 & = 0 \\ 5 x (x + 8) + (x + 8) & = 0 \\ (x + 8) (5 x + 1) & = 0 \\ x + 8 & = 0 \\ x_{1} & = - 8 \\ 5 x + 1 & = 0 \\ 5 x & = - 1 \\ x_{2} & = - \frac{1}{5} \\ 5 x^{2} + 41 x + 8 & = 5 (x + 8) (x + \frac{1}{5}) \\ 5 x^{2} + 41 x + 8 & = 5 (x + 8) (\frac{5 x + 1}{5}) \\ 5 x^{2} + 41 x + 8 & = (x + 8) (5 x + 1) \end{matrix}$
$\begin{matrix} 6 x^{2} + 7 x - 10 \\ 6 x^{2} + 7 x - 10 & = 0 \\ 6 x^{2} + 12 x - 5 x - 10 & = 0 \\ 6 x (x + 2) - 5 (x + 2) & = 0 \\ (x + 2) (6 x - 5) & = 0 \\ x + 2 & = 0 \\ x_{1} & = - 2 \\ 6 x - 5 & = 0 \\ 6 x & = 5 \\ x_{2} & = \frac{5}{6} \\ 6 x^{2} + 7 x - 10 & = 6 (x + 2) (x - \frac{5}{6}) \\ 6 x^{2} + 7 x - 10 & = 6 (x + 2) (\frac{6 x - 5}{6}) \\ 6 x^{2} + 7 x - 10 & = (x + 2) (6 x - 5) \end{matrix}$
$\begin{matrix} 12 x^{2} - 25 x + 12 \\ 12 x^{2} - 25 x + 12 & = 0 \\ 12 x^{2} - 16 x - 9 x + 12 & = 0 \\ 4 x (3 x - 4) - 3 (3 x - 4) & = 0 \\ (3 x - 4) (4 x - 3) & = 0 \\ 3 x - 4 & = 0 \\ 3 x & = 4 \\ x_{1} & = \frac{4}{3} \\ 4 x - 3 & = 0 \\ 4 x & = 3 \\ x_{2} & = \frac{3}{4} \\ 12 x^{2} - 25 x + 12 & = 12 (x - \frac{4}{3}) (x - \frac{3}{4}) \\ 12 x^{2} - 25 x + 12 & = 12 (\frac{3 x - 4}{3}) (\frac{4 x - 3}{4}) \\ 12 x^{2} - 25 x + 12 & = (3 x - 4) (4 x - 3) \end{matrix}$
$\begin{matrix} 8 x^{2} + 50 x + 63 \\ 8 x^{2} + 50 x + 63 & = 0 \\ x & = \frac{- 50 \pm \sqrt{5 0^{2} - 4 (8) (63)}}{2 (8)} \\ x & = \frac{- 50 \pm \sqrt{2500 - 2016}}{16} \\ x & = \frac{- 50 \pm \sqrt{484}}{16} \\ x & = \frac{- 50 \pm 22}{16} \\ x_{1} & = \frac{- 50 + 22}{16} = - \frac{}{} = - \frac{7}{4} \\ x_{2} & = \frac{- 50 - 22}{16} = - \frac{}{} = - \frac{9}{2} \\ 8 x^{2} + 50 x + 63 & = 8 (x + \frac{7}{4}) (x + \frac{9}{2}) \\ 8 x^{2} + 50 x + 63 & = 8 (\frac{4 x + 7}{4}) (\frac{2 x + 9}{2}) \\ 8 x^{2} + 50 x + 63 & = (4 x + 7) (2 x + 9) \end{matrix}$
$\begin{matrix} 27 x^{2} + 30 x + 7 \\ 27 x^{2} + 30 x + 7 & = 0 \\ 27 x^{2} + 9 x + 21 x + 7 & = 0 \\ 9 x (3 x + 1) + 7 (3 x + 1) & = 0 \\ (3 x + 1) (9 x + 7) & = 0 \\ 3 x + 1 & = 0 \\ 3 x & = - 1 \\ x_{1} & = - \frac{1}{3} \\ 9 x + 7 & = 0 \\ 9 x & = - 7 \\ x_{2} & = - \frac{7}{9} \\ 27 x^{2} + 30 x + 7 & = 27 (x + \frac{1}{3}) (x + \frac{7}{9}) \\ 27 x^{2} + 30 x + 7 & = 27 (\frac{3 x + 1}{3}) (\frac{9 x + 7}{9}) \\ 27 x^{2} + 30 x + 7 & = (3 x + 1) (9 x + 7) \end{matrix}$
$\begin{matrix} 30 x^{2} - 61 x + 30 \\ 30 x^{2} - 61 x + 30 & = 0 \\ 30 x^{2} - 36 x - 25 x + 30 & = 0 \\ 6 x (5 x - 6) - 5 (5 x - 6) & = 0 \\ (5 x - 6) (6 x - 5) & = 0 \\ 5 x - 6 & = 0 \\ 5 x & = 6 \\ x_{1} & = \frac{6}{5} \\ 6 x - 5 & = 0 \\ 6 x & = 5 \\ x_{2} & = \frac{5}{6} \\ 30 x^{2} - 61 x + 30 & = 30 (x - \frac{6}{5}) (x - \frac{5}{6}) \\ 30 x^{2} - 61 x + 30 & = 30 (\frac{5 x - 6}{5}) (\frac{6 x - 5}{6}) \\ 30 x^{2} - 61 x + 30 & = (5 x - 6) (6 x - 5) \end{matrix}$
$\begin{matrix} 11 x^{2} - 153 x - 180 \\ 11 x^{2} - 153 x - 180 & = 0 \\ 11 x^{2} - 165 x + 12 x - 180 & = 0 \\ 11 x (x - 15) + 12 (x - 15) & = 0 \\ (x - 15) (11 x + 12) & = 0 \\ x - 15 & = 0 \\ x_{1} & = 15 \\ 11 x + 12 & = 0 \\ 11 x & = - 12 \\ x & = - \frac{12}{11} \\ 11 x^{2} - 153 x - 180 & = 11 (x - 15) (x + \frac{12}{11}) \\ 11 x^{2} - 153 x - 180 & = 11 (x - 15) (\frac{11 x + 12}{11}) \\ 11 x^{2} - 153 x - 180 & = (x - 15) (11 x + 12) \end{matrix}$
$\begin{matrix} 6 - x - x^{2} \\ x^{2} + x - 6 & = 0 \\ (x + 3) (x - 2) & = 0 \\ x + 3 & = 0 \\ x_{1} & = - 3 \\ x - 2 & = 0 \\ x_{2} & = 2 \\ x^{2} + x - 6 & = (x + 3) (x - 2) \end{matrix}$
$\begin{matrix} 5 - 9 x - 2 x^{2} \\ 2 x^{2} + 9 x - 5 & = 0 \\ 2 x^{2} - x + 10 x - 5 & = 0 \\ x (2 x - 1) + 5 (2 x - 1) & = 0 \\ (2 x - 1) (x + 5) & = 0 \\ 2 x - 1 & = 0 \\ 2 x & = 1 \\ x_{1} & = \frac{1}{2} \\ x + 5 & = 0 \\ x_{2} & = - 5 \\ 2 x^{2} + 9 x - 5 & = 2 (x - \frac{1}{2}) (x + 5) \\ 2 x^{2} + 9 x - 5 & = 2 (\frac{2 x - 1}{2}) (x + 5) \\ 2 x^{2} + 9 x - 5 & = (2 x - 1) (x + 5) \end{matrix}$
$\begin{matrix} 15 + 4 x - 4 x^{2} \\ 4 x^{2} - 4 x - 15 & = 0 \\ x & = \frac{- (- 4) \pm \sqrt{{(- 4)}^{2} - 4 (4) (- 15)}}{2 (4)} \\ x & = \frac{4 \pm \sqrt{16 + 240}}{8} \\ x & = \frac{4 \pm \sqrt{256}}{8} \\ x & = \frac{4 \pm 16}{8} \\ x_{1} & = \frac{4 + 16}{8} = \frac{}{} = \frac{5}{2} \\ x_{2} & = \frac{4 - 16}{8} = - \frac{}{} = - \frac{3}{2} \\ 4 x^{2} - 4 x - 15 & = 4 (x - \frac{5}{2}) (x + \frac{3}{2}) \\ 4 x^{2} - 4 x - 15 & = 4 (\frac{2 x - 5}{2}) (\frac{2 x + 3}{2}) \\ 4 x^{2} - 4 x - 15 & = (2 x - 5) (2 x + 3) \end{matrix}$
$\begin{matrix} 4 - 13 x - 12 x^{2} \\ 12 x^{2} + 13 x - 4 & = 0 \\ 12 x^{2} - 3 x + 16 x - 4 & = 0 \\ 3 x (4 x - 1) + 4 (4 x - 1) & = 0 \\ (4 x - 1) (3 x + 4) & = 0 \\ 4 x - 1 & = 0 \\ 4 x & = 1 \\ x_{1} & = \frac{1}{4} \\ 3 x + 4 & = 0 \\ 3 x & = - 4 \\ x_{2} & = - \frac{4}{3} \\ 12 x^{2} + 13 x - 4 & = 12 (x - \frac{1}{4}) (x + \frac{4}{3}) \\ 12 x^{2} + 13 x - 4 & = 12 (\frac{4 x - 1}{4}) (\frac{3 x + 4}{3}) \\ 12 x^{2} + 13 x - 4 & = (4 x - 1) (3 x + 4) \end{matrix}$
$\begin{matrix} 72 x^{2} - 55 x - 7 \\ 72 x^{2} - 55 x - 7 & = 0 \\ 72 x^{2} + 8 x - 63 x - 7 & = 0 \\ 8 x (9 x + 1) - 7 (9 x + 1) & = 0 \\ (9 x + 1) (8 x - 7) & = 0 \\ 9 x + 1 & = 0 \\ 9 x & = - 1 \\ x_{1} & = - \frac{1}{9} \\ 8 x - 7 & = 0 \\ 8 x & = 7 \\ x & = \frac{7}{8} \\ 72 x^{2} - 55 x - 7 & = 72 (x + \frac{1}{9}) (x - \frac{7}{8}) \\ 72 x^{2} - 55 x - 7 & = 72 (\frac{9 x + 1}{9}) (\frac{8 x - 7}{8}) \\ 72 x^{2} - 55 x - 7 & = (9 x + 1) (8 x - 7) \end{matrix}$
$\begin{matrix} 6 + 31 x - 30 x^{2} \\ 30 x^{2} - 31 x - 6 & = 0 \\ 30 x^{2} + 5 x - 36 x - 6 & = 0 \\ 5 x (6 x + 1) - 6 (6 x + 1) & = 0 \\ (6 x + 1) (5 x - 6) & = 0 \\ 6 x + 1 & = 0 \\ 6 x & = - 1 \\ x_{1} & = - \frac{1}{6} \\ 5 x - 6 & = 0 \\ 5 x & = 6 \\ x_{2} & = \frac{6}{5} \\ 30 x^{2} - 31 x - 6 & = 30 (x + \frac{1}{6}) (x - \frac{6}{5}) \\ 30 x^{2} - 31 x - 6 & = 30 (\frac{6 x + 1}{6}) (\frac{5 x - 6}{5}) \\ 30 x^{2} - 31 x - 6 & = (6 x + 1) (5 x - 6) \end{matrix}$
$\begin{matrix} 10 x^{2} + 207 x - 63 \\ x & = \frac{- 207 \pm \sqrt{20 7^{2} - 4 (10) (- 63)}}{2 (10)} \\ x & = \frac{- 207 \pm \sqrt{42849 + 2520}}{20} \\ x & = \frac{- 207 \pm \sqrt{45369}}{20} \\ x & = \frac{- 207 \pm 213}{20} \\ x_{1} & = \frac{- 207 + 213}{20} = \frac{}{} = \frac{3}{10} \\ x_{2} & = \frac{- 207 - 213}{20} = - \frac{}{20} = - 24 \\ 10 x^{2} + 207 x - 63 & = 10 (x - \frac{3}{10}) (x + 24) \\ 10 x^{2} + 207 x - 63 & = 10 (\frac{10 x - 3}{10}) (x - 24) \\ 10 x^{2} + 207 x - 63 & = (10 x - 3) (x - 24) \end{matrix}$
$\begin{matrix} 100 - 15 x - x^{2} \\ x^{2} + 15 x - 100 & = 0 \\ (x + 20) (x - 5) & = 0 \\ x + 20 & = 0 \\ x_{1} & = - 20 \\ x - 5 & = 0 \\ x_{2} & = 5 \\ x^{2} + 15 x - 100 & = (x + 20) (x - 5) \end{matrix}$
$\begin{matrix} 18 x^{2} + 31 x - 49 \\ 18 x^{2} + 31 x - 49 & = 0 \\ 18 x^{2} - 18 x + 49 x - 49 & = 0 \\ 18 x (x - 1) + 49 (x - 1) & = 0 \\ (x - 1) (18 x + 49) & = 0 \\ x - 1 & = 0 \\ x_{1} & = 1 \\ 18 x + 49 & = 0 \\ 18 x & = - 49 \\ x_{2} & = - \frac{49}{18} \\ 18 x^{2} + 31 x - 49 & = 18 (x - 1) (x + \frac{49}{18}) \\ 18 x^{2} + 31 x - 49 & = 18 (x - 1) (\frac{18 x + 49}{18}) \\ 18 x^{2} + 31 x - 49 & = (x - 1) (18 x + 49) \end{matrix}$
$\begin{matrix} 6 x^{2} - a x - 2 a^{2} \\ 6 x^{2} - a x - 2 a^{2} & = 0 \\ 6 x^{2} + 3 a x - 4 a x - 2 a^{2} & = 0 \\ 3 x (2 x + a) - 2 a (2 x + a) & = 0 \\ (2 x + a) (3 x - 2 a) & = 0 \\ 2 x + a & = 0 \\ 2 x & = - a \\ x_{1} & = - \frac{a}{2} \\ 3 x - 2 a & = 0 \\ 3 x & = 2 a \\ x_{2} & = \frac{2 a}{3} \\ 6 x^{2} - a x - 2 a^{2} & = 6 (x + \frac{a}{2}) (x - \frac{2 a}{3}) \\ 6 x^{2} - a x - 2 a^{2} & = 6 (\frac{2 x + a}{2}) (\frac{3 x - 2 a}{3}) \\ 6 x^{2} - a x - 2 a^{2} & = (2 x + a) (3 x - 2 a) \end{matrix}$
$\begin{matrix} 5 x^{2} + 22 x y - 15 y^{2} \\ 5 x^{2} + 22 x y - 15 y^{2} & = 0 \\ 5 x^{2} + 25 x y - 3 x y - 15 y^{2} & = 0 \\ 5 x (x + 5 y) - 3 y (x + 5 y) & = 0 \\ (x + 5 y) (5 x - 3 y) & = 0 \\ x + 5 y & = 0 \\ x_{1} & = - 5 y \\ 5 x - 3 y & = 0 \\ 5 x & = 3 y \\ x_{2} & = \frac{3 y}{5} \\ 5 x^{2} + 22 x y - 15 y^{2} & = 5 (x + 5 y) (x - \frac{3 y}{5}) \\ 5 x^{2} + 22 x y - 15 y^{2} & = 5 (x + 5 y) (\frac{5 x - 3 y}{5}) \\ 5 x^{2} + 22 x y - 15 y^{2} & = (x + 5 y) (5 x - 3 y) \end{matrix}$
$\begin{matrix} 15 x^{2} - 32 m x - 7 m^{2} \\ 15 x^{2} - 32 m x - 7 m^{2} & = 0 \\ 15 x^{2} + 3 m x - 35 m x - 7 m^{2} & = 0 \\ 3 x (5 x + m) - 7 m (5 x + m) & = 0 \\ (5 x + m) (3 x - 7 m) & = 0 \\ 5 x + m & = 0 \\ 5 x & = - m \\ x_{1} & = - \frac{m}{5} \\ 3 x - 7 m & = 0 \\ 3 x & = 7 m \\ x_{2} & = \frac{7 m}{3} \\ 15 x^{2} - 32 m x - 7 m^{2} & = 15 (x + \frac{m}{5}) (x - \frac{7 m}{3}) \\ 15 x^{2} - 32 m x - 7 m^{2} & = 15 (\frac{5 x + m}{5}) (\frac{3 x - 7 m}{3}) \\ 15 x^{2} - 32 m x - 7 m^{2} & = (5 x + m) (3 x - 7 m) \end{matrix}$