Ejercicio 265 – Solucionario Baldor

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CAPITULO XXXIII

Ecuaciones de segundo grado

Ejercicio 265

Resolver las siguientes ecuaciones por la fórmula general:

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

$\begin{matrix} 3 x^{2} - 5 x + 2 & = 0 \\ x & = \frac{5 \pm \sqrt{5^{2} - 4 (3) (2)}}{2 (3)} \\ x & = \frac{5 \pm \sqrt{25 - 24}}{6} \\ x & = \frac{5 \pm 1}{6} \\ x_{1} & = \frac{6}{6} \\ x_{1} & = 1 \\ x_{2} & = \frac{}{} \\ x_{2} & = \frac{2}{3} \end{matrix}$
$\begin{matrix} 4 x^{2} + 3 x - 22 & = 0 \\ x & = \frac{- 3 \pm \sqrt{3^{2} - 4 (4) (- 22)}}{2 (4)} \\ x & = \frac{- 3 \pm \sqrt{9 - 352}}{8} \\ x & = \frac{- 3 \pm \sqrt{361}}{8} \\ x & = \frac{- 3 \pm 19}{8} \\ x_{1} & = \frac{}{8} \\ x_{1} & = 2 \\ x_{2} & = \frac{-}{} \\ x_{2} & = - \frac{11}{4} \end{matrix}$
$\begin{matrix} x^{2} + 11 x & = - 24 \\ x^{2} + 11 x + 24 & = 0 \\ x & = \frac{- 11 \pm \sqrt{1 1^{2} - 4 (1) (24)}}{2 (1)} \\ x & = \frac{- 11 \pm \sqrt{121 - 96}}{2} \\ x & = \frac{- 11 \pm \sqrt{25}}{2} \\ x & = \frac{- 11 \pm 5}{2} \\ x_{1} & = \frac{-}{2} \\ x_{1} & = - 8 \\ x_{2} & = \frac{-}{2} \\ x_{2} & = - 3 \end{matrix}$
$\begin{matrix} x^{2} & = 16 x - 63 \\ x^{2} - 16 x + 63 & = 0 \\ x & = \frac{16 \pm \sqrt{1 6^{2} - 4 (1) (63)}}{2 (1)} \\ x & = \frac{16 \pm \sqrt{1 6^{2} - 4 (1) (63)}}{2 (1)} \\ x & = \frac{16 \pm \sqrt{256 - 252}}{2} \\ x & = \frac{16 \pm \sqrt{4}}{2} \\ x & = \frac{16 \pm 2}{2} \\ x_{1} & = \frac{}{2} \\ x_{1} & = 9 \\ x_{2} & = \frac{}{2} \\ x_{2} & = 7 \end{matrix}$
$\begin{matrix} 12 x - 4 - 9 x^{2} & = 0 \\ 9 x^{2} - 12 x + 4 & = 0 \\ x & = \frac{12 \pm \sqrt{1 2^{2} - 4 (9) (4)}}{2 (9)} \\ x & = \frac{12 \pm}{18} \\ x & = \frac{}{} \\ x & = \frac{2}{3} \end{matrix}$
$\begin{matrix} 5 x^{2} - 7 x - 90 & = 0 \\ x & = \frac{7 \pm \sqrt{7^{2} - 4 (5) (- 90)}}{2 (5)} \\ x & = \frac{7 \pm \sqrt{49 + 1800}}{10} \\ x & = \frac{7 \pm \sqrt{1849}}{10} \\ x & = \frac{7 \pm 43}{10} \\ x_{1} & = \frac{50}{10} \\ x_{1} & = 5 \\ x_{2} & = \frac{-}{} \\ x_{2} & = - \frac{18}{5} \end{matrix}$
$\begin{matrix} 6 x^{2} & = x + 222 \\ 6 x^{2} - x - 222 & = 0 \\ x & = \frac{1 \pm \sqrt{1 - 4 (6) (- 222)}}{2 (6)} \\ x & = \frac{1 \pm \sqrt{1 + 5328}}{12} \\ x & = \frac{1 \pm 73}{12} \\ x_{1} & = \frac{}{} \\ x_{1} & = \frac{37}{6} \\ x_{2} & = \frac{-}{12} \\ x_{2} & = - 6 \end{matrix}$
$\begin{matrix} x + 11 & = 10 x^{2} \\ 10 x^{2} - x - 11 & = 0 \\ x & = \frac{1 \pm \sqrt{1 - 4 (10) (- 11)}}{2 (10)} \\ x & = \frac{1 \pm \sqrt{1 + 440}}{20} \\ x & = \frac{1 \pm 21}{20} \\ x_{1} & = \frac{}{} \\ x_{1} & = \frac{11}{10} \\ x_{2} & = \frac{- 20}{20} \\ x_{2} & = - 1 \end{matrix}$
$\begin{matrix} 49 x^{2} - 70 x + 25 & = 0 \\ x & = \frac{70 \pm \sqrt{7 0^{2} - 4 (49) (25)}}{2 (49)} \\ x & = \frac{70 \pm}{98} \\ x & = \frac{}{} \\ x & = \frac{5}{7} \end{matrix}$
$\begin{matrix} 12 x - 7 x^{2} + 64 & = 0 \\ 7 x^{2} - 12 x - 64 & = 0 \\ x & = \frac{12 \pm \sqrt{1 2^{2} - 4 (7) (- 64)}}{2 (7)} \\ x & = \frac{12 \pm \sqrt{144 + 1792}}{14} \\ x & = \frac{12 \pm \sqrt{1936}}{14} \\ x & = \frac{12 \pm 44}{14} \\ x_{1} & = \frac{}{14} \\ x_{1} & = 4 \\ x_{2} & = \frac{-}{} \\ x_{2} & = - \frac{16}{7} \end{matrix}$
$\begin{matrix} x^{2} & = - 15 x - 56 \\ x^{2} + 15 x + 56 & = 0 \\ x & = \frac{- 15 \pm \sqrt{1 5^{2} - 4 (1) (56)}}{2 (1)} \\ x & = \frac{- 15 \pm \sqrt{225 - 224}}{2} \\ x & = \frac{- 15 \pm 1}{2} \\ x_{1} & = \frac{-}{2} \\ x_{1} & = - 7 \\ x_{2} & = \frac{-}{2} \\ x_{2} & = - 8 \end{matrix}$
$\begin{matrix} 32 x^{2} + 18 x - 17 & = 0 \\ x & = \frac{- 18 \pm \sqrt{1 8^{2} - 4 (32) (- 17)}}{2 (32)} \\ x & = \frac{- 18 \pm \sqrt{324 + 2176}}{64} \\ x & = \frac{- 18 \pm \sqrt{2500}}{64} \\ x & = \frac{- 18 \pm 50}{64} \\ x_{1} & = \frac{32}{} \\ x_{1} & = \frac{1}{2} \\ x_{2} & = \frac{-}{} \\ x_{2} & = - \frac{17}{16} \end{matrix}$
$\begin{matrix} 176 x & = 121 + 64 x^{2} \\ 64 x^{2} - 176 x + 121 & = 0 \\ x & = \frac{- 176 \pm \sqrt{17 6^{2} - 4 (64) (121)}}{2 (64)} \\ x & = \frac{- 176 \pm}{128} \\ x & = - \frac{}{} \\ x & = - \frac{11}{8} \end{matrix}$
$\begin{matrix} 8 x + 5 & = 36 x^{2} \\ 36 x^{2} - 8 x - 5 & = 0 \\ x & = \frac{8 \pm \sqrt{8^{2} - 4 (36) (- 5)}}{2 (36)} \\ x & = \frac{8 \pm \sqrt{64 + 720}}{72} \\ x & = \frac{8 \pm \sqrt{784}}{72} \\ x & = \frac{8 \pm 28}{72} \\ x_{1} & = \frac{36}{} \\ x_{1} & = \frac{1}{2} \\ x_{2} & = - \frac{}{} \\ x_{2} & = - \frac{5}{18} \end{matrix}$
$\begin{matrix} 27 x^{2} + 12 x - 7 & = 0 \\ x & = \frac{- 12 \pm \sqrt{1 2^{2} - 4 (27) (- 7)}}{2 (27)} \\ x & = \frac{- 12 \pm \sqrt{144 + 756}}{54} \\ x & = \frac{- 12 \pm \sqrt{900}}{54} \\ x & = \frac{- 12 \pm 30}{54} \\ x_{1} & = \frac{18}{} \\ x_{1} & = \frac{1}{3} \\ x_{2} & = - \frac{}{} \\ x_{2} & = - \frac{7}{9} \end{matrix}$
$\begin{matrix} 15 x & = 25 x^{2} + 2 \\ 25 x^{2} - 15 x + 2 & = 0 \\ x & = \frac{15 \pm \sqrt{1 5^{2} - 4 (25) (2)}}{2 (25)} \\ x & = \frac{15 \pm \sqrt{225 - 200}}{50} \\ x & = \frac{15 \pm \sqrt{25}}{50} \\ x & = \frac{15 \pm 5}{50} \\ x_{1} & = \frac{20}{50} \\ x_{1} & = \frac{2}{5} \\ x_{2} & = \frac{10}{50} \\ x_{2} & = \frac{1}{5} \end{matrix}$
$\begin{matrix} 8 x^{2} - 2 x - 3 & = 0 \\ x & = \frac{2 \pm \sqrt{2^{2} - 4 (8) (- 3)}}{2 (8)} \\ x & = \frac{2 \pm \sqrt{4 + 96}}{16} \\ x & = \frac{2 \pm \sqrt{100}}{16} \\ x & = \frac{2 \pm 10}{16} \\ x_{1} & = \frac{}{} \\ x_{1} & = \frac{3}{4} \\ x_{2} & = \frac{- 8}{} \\ x_{2} & = - \frac{1}{2} \end{matrix}$
$\begin{matrix} 105 & = x + 2 x^{2} \\ 2 x^{2} + x - 105 & = 0 \\ x & = \frac{- 1 \pm \sqrt{1 - 4 (2) (- 105)}}{2 (2)} \\ x & = \frac{- 1 \pm \sqrt{1 - 4 (2) (- 105)}}{4} \\ x & = \frac{- 1 \pm \sqrt{1 + 840}}{4} \\ x & = \frac{- 1 \pm \sqrt{841}}{4} \\ x & = \frac{- 1 \pm 29}{4} \\ x_{1} & = \frac{}{4} \\ x_{1} & = 7 \\ x_{2} & = \frac{-}{} \\ x_{2} & = - \frac{15}{2} \end{matrix}$