▶️ Ejercicio 266 álgebra - Solucionario Baldor ✅✅

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CAPITULO XXXIII

Ecuaciones de segundo grado

Ejercicio 266

Resolver las siguientes ecuaciones llevándolas a la forma

a x^{2} + b x + c = 0

y aplicando la fórmula general:

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

$\begin{matrix} x (x + 3) & = 5 x + 3 \\ x^{2} + 3 x & = 5 x + 3 \\ x^{2} + 3 x - 5 x - 3 & = 0 \\ x^{2} - 2 x - 3 & = 0 \\ x & = \frac{2 \pm \sqrt{2^{2} - 4 (1) (- 3)}}{2 (1)} \\ x & = \frac{2 \pm \sqrt{4 + 12}}{2} \\ x & = \frac{2 \pm \sqrt{16}}{2} \\ x & = \frac{2 \pm 4}{2} \\ x_{1} & = \frac{}{2} \\ x_{1} & = 3 \\ x_{2} & = \frac{- 2}{2} \\ x_{2} & = - 1 \end{matrix}$
$\begin{matrix} 3 (3 x - 2) & = (x + 4) (4 - x) \\ 9 x - 6 & = 4 x - x^{2} + 16 - 4 x \\ 9 x - 6 - 16 + x^{2} & = 0 \\ x^{2} + 9 x - 22 & = 0 \\ x & = \frac{- 9 \pm \sqrt{9^{2} - 4 (1) (- 22)}}{2 (1)} \\ x & = \frac{- 9 \pm \sqrt{81 + 88}}{2} \\ x & = \frac{- 9 \pm \sqrt{169}}{2} \\ x & = \frac{- 3 \pm 13}{2} \\ x_{1} & = \frac{}{2} \\ x_{1} & = 5 \\ x_{2} & = \frac{-}{2} \\ x_{2} & = - 4 \end{matrix}$
$\begin{matrix} 9 x + 1 & = 3 (x^{2} - 5) - (x - 3) (x + 2) \\ 9 x + 1 & = 3 x^{2} - 15 - (x^{2} - x - 6) \\ 9 x + 1 & = 3 x^{2} - 15 - x^{2} + x + 6 \\ 0 & = 2 x^{2} - 9 + x - 9 x - 1 \\ 0 & = 2 x^{2} - 8 x - 10 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 2 \\ x^{2} - 4 x - 5 & = 0 \\ x & = \frac{- 4 \pm \sqrt{4^{2} - 4 (1) (- 5)}}{2 (1)} \\ x & = \frac{- 4 \pm \sqrt{16 + 20}}{2} \\ x & = \frac{- 4 \pm \sqrt{36}}{2} \\ x & = \frac{- 4 \pm 6}{2} \\ x_{1} & = \frac{2}{2} \\ x_{1} & = 1 \\ x_{2} & = \frac{-}{2} \\ x_{2} & = - 5 \end{matrix}$
$\begin{matrix} {(2 x - 3)}^{2} - {(x + 5)}^{2} & = - 23 \\ 4 x^{2} - 12 x + 9 - (x^{2} + 10 x + 25) & = - 23 \\ 4 x^{2} - 12 x + 9 - x^{2} - 10 x - 25 & = - 23 \\ 3 x^{2} - 22 x - 16 + 23 & = 0 \\ 3 x^{2} - 22 x + 7 & = 0 \\ x & = \frac{22 \pm \sqrt{2 2^{2} - 4 (3) (7)}}{2 (3)} \\ x & = \frac{22 \pm \sqrt{484 - 84}}{6} \\ x & = \frac{22 \pm \sqrt{400}}{6} \\ x & = \frac{22 \pm 20}{6} \\ x_{1} & = \frac{2}{} \\ x_{1} & = \frac{1}{3} \\ x_{2} & = \frac{}{6} \\ x_{2} & = 7 \end{matrix}$
$\begin{matrix} 25 {(x + 2)}^{2} & = {(x - 7)}^{2} - 81 \\ 25 (x^{2} + 4 x + 4) & = x^{2} - 14 x + 49 - 81 \\ 25 x^{2} + 100 x + 100 & = x^{2} - 14 x - 32 \\ 25 x^{2} + 100 x + 100 - x^{2} + 14 x + 32 & = 0 \\ 24 x^{2} + 114 x + 132 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 6 \\ 4 x^{2} + 19 x + 22 & = 0 \\ x & = \frac{- 19 \pm \sqrt{1 9^{2} - 4 (4) (22)}}{2 (4)} \\ x & = \frac{- 19 \pm \sqrt{361 - 352}}{8} \\ x & = \frac{- 19 \pm \sqrt{9}}{8} \\ x & = \frac{- 19 \pm 3}{8} \\ x_{1} & = \frac{-}{} \\ x_{1} & = - \frac{11}{4} \\ x_{2} & = \frac{-}{8} \\ x_{2} & = - 2 \end{matrix}$
$\begin{matrix} 3 x (x - 2) - (x - 6) & = 23 (x - 3) \\ 3 x^{2} - 6 x - x + 6 & = 23 x - 69 \\ 3 x^{2} - 7 x + 6 - 23 x + 69 & = 0 \\ 3 x^{2} - 30 x + 75 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 3 \\ x^{2} - 10 x + 25 & = 0 \\ x & = \frac{10 \pm \sqrt{1 0^{2} - 4 (1) (25)}}{2 (1)} \\ x & = \frac{10 \pm}{2} \\ x & = \frac{}{2} \\ x & = 5 \end{matrix}$
$\begin{matrix} 7 (x - 3) - 5 (x^{2} - 1) & = x^{2} - 5 (x + 2) \\ 7 x - 21 - 5 x^{2} + 5 & = x^{2} - 5 x - 10 \\ 7 x - 16 - 5 x^{2} - x^{2} + 5 x + 10 & = 0 \\ - 6 x^{2} + 12 x - 6 & = 0 \\ D i v i d i e n d o p a r a - 6 l a e c u a c i ó n \\ x^{2} - 2 x + 1 & = 0 \\ x & = \frac{2 \pm \sqrt{2^{2} - 4 (1) (1)}}{2 (1)} \\ x & = \frac{2 \pm}{2} \\ x & = \frac{2}{2} \\ x & = 1 \end{matrix}$
$\begin{matrix} {(x - 5)}^{2} - {(x - 6)}^{2} & = {(2 x - 3)}^{2} - 118 \\ x^{2} - 10 x + 25 - (x^{2} - 12 x + 36) & = 4 x^{2} - 12 x + 9 - 118 \\ x^{2} - 10 x + 25 - x^{2} + 12 x - 36 & = 4 x^{2} - 12 x - 109 \\ 0 & = 4 x^{2} - 12 x - 109 - 2 x + 11 \\ 0 & = 4 x^{2} - 14 x - 98 \\ D i v i d i m o s l a e c u a c i ó n p a r a 2 \\ 2 x^{2} - 7 x - 49 & = 0 \\ x & = \frac{7 \pm \sqrt{7^{2} - 4 (2) (- 49)}}{2 (2)} \\ x & = \frac{7 \pm \sqrt{49 + 392}}{4} \\ x & = \frac{7 \pm \sqrt{441}}{4} \\ x & = \frac{7 \pm 21}{4} \\ x_{1} & = \frac{}{4} \\ x_{1} & = 7 \\ x_{2} & = \frac{-}{} \\ x_{2} & = - \frac{7}{2} \end{matrix}$
$\begin{matrix} {(5 x - 2)}^{2} - {(3 x + 1)}^{2} - x^{2} - 60 & = 0 \\ 25 x^{2} - 20 x + 4 - (9 x^{2} + 6 x + 1) - x^{2} - 60 & = 0 \\ 25 x^{2} - 20 x - 9 x^{2} - 6 x - 1 - x^{2} - 56 & = 0 \\ 15 x^{2} - 26 x - 57 & = 0 \\ x & = \frac{26 \pm \sqrt{2 6^{2} - 4 (15) (- 57)}}{2 (15)} \\ x & = \frac{26 \pm \sqrt{676 + 3420}}{30} \\ x & = \frac{26 \pm \sqrt{4096}}{30} \\ x & = \frac{26 \pm 64}{30} \\ x_{1} & = \frac{}{30} \\ x_{1} & = 3 \\ x_{2} & = \frac{-}{} \\ x_{2} & = - \frac{19}{15} \end{matrix}$
$\begin{matrix} {(x + 4)}^{3} - {(x - 3)}^{3} & = 343 \\ x^{3} + 12 x^{2} + 48 x + 64 - (x^{3} - 9 x^{2} + 27 x - 27) & = 343 \\ x^{3} + 12 x^{2} + 48 x + 64 - x^{3} + 9 x^{2} - 27 x + 27 - 343 & = 0 \\ 21 x^{2} + 21 x - 252 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 21 \\ x^{2} + x - 12 & = 0 \\ x & = \frac{- 1 \pm \sqrt{1 - 4 (1) (- 12)}}{2 (1)} \\ x & = \frac{- 1 \pm \sqrt{1 + 48}}{2} \\ x & = \frac{- 1 \pm \sqrt{49}}{2} \\ x & = \frac{- 1 \pm 7}{2} \\ x_{1} & = \frac{-}{2} \\ x_{1} & = - 4 \\ x_{2} & = \frac{}{2} \\ x_{2} & = 3 \end{matrix}$
$\begin{matrix} {(x + 2)}^{3} - {(x - 1)}^{3} & = x (3 x + 4) + 8 \\ x^{3} + 6 x^{2} + 12 x + 8 - (x^{3} - 3 x^{2} + 3 x - 1) & = 3 x^{2} + 4 x + 8 \\ x^{3} + 6 x^{2} + 12 x + 8 - x^{3} + 3 x^{2} - 3 x + 1 & = 3 x^{2} + 4 x + 8 \\ 6 x^{2} + 9 x + 1 - 4 x & = 0 \\ 6 x^{2} + 5 x + 1 & = 0 \\ x & = \frac{- 5 \pm \sqrt{5^{2} - 4 (6) (1)}}{2 (6)} \\ x & = \frac{- 5 \pm}{12} \\ x & = \frac{- 5 \pm 1}{12} \\ x_{1} & = \frac{- 4}{} \\ x_{1} & = - \frac{1}{3} \\ x_{2} & = \frac{- 6}{} \\ x_{2} & = - \frac{1}{2} \end{matrix}$
$\begin{matrix} {(5 x - 4)}^{2} - (3 x + 5) (2 x - 1) & = 20 x (x - 2) + 27 \\ 25 x^{2} - 40 x + 16 - (6 x^{2} - 3 x + 10 x - 5) & = 20 x^{2} - 40 x + 27 \\ 25 x^{2} + 16 - 6 x^{2} - 7 x + 5 - 20 x^{2} - 27 & = 0 \\ - x^{2} - 7 x - 6 & = 0 \\ x^{2} + 7 x + 6 & = 0 \\ x & = \frac{- 7 \pm \sqrt{7^{2} - 4 (1) (6)}}{2 (1)} \\ x & = \frac{- 7 \pm \sqrt{49 - 24}}{2} \\ x & = \frac{- 7 \pm \sqrt{25}}{2} \\ x & = \frac{- 7 \pm 5}{2} \\ x_{1} & = \frac{- 2}{2} \\ x_{1} & = - 1 \\ x_{2} & = \frac{-}{2} \\ x_{2} & = - 6 \end{matrix}$

Categories: Capítulo XXXIII