Ejercicios Resueltos de Ecuaciones

a)  $x/2+1=x/3+6$→$x/3-x/2=1-6$
→$\displaystyle\frac{2x}{6}-\frac{3x}{6}=-5$→$-\displaystyle\frac{x}{6}=-5$→$x=30\bullet$

b)  $\displaystyle\frac{3(x-1)+x}{15}-\displaystyle\frac{1-x}{12}=\left(\displaystyle\frac{5}{3}\right)^{-1}(x-4)$
→$\displaystyle\frac{3x-3+x}{3.5}-\displaystyle\frac{1-x}{3.4}=\displaystyle\frac{3x-12}{5}$
→$\displaystyle\frac{4(4x-3)}{60}-\displaystyle\frac{5(1-x)}{60}=\displaystyle\frac{12(3x-12)}{60}$
→$16x-12-5+5x=36x-144$→$15x=127$→$x=127/15\bullet$

c)  $x^2+6x=0$→$a=1$  y  $b=6$ sumo a ambos miembros $(b/2a)^2$
→$x^2+6x+(6/2)^2=(6/2)^2$→$(x+3)^2=9$
→$x+3=\pm3$→$x_1+3=3$→$x_1=0\bullet$→$x_2+3=-3$→$x_2=-6\bullet$

d)  $x^2-4x+3=0$→$(x-1)(x-3)$→$x_1=1\bullet$→$x_2=3\bullet$

e)  $x^2-4x-12=0$→$(x-6)(x+2)$→$x_1=6\bullet$→$x_2=-2\bullet$

f)  $(x+1)^2=(1-3x)^2$→$x^2+2x+1=1-6x+9x^2$
→$8x^2-8x=0$→$8x(x-1)=0$→$x_1=0\bullet$→$x_2=1\bullet$

g)  $5x^2-19x+4=0$→$x^2-(19/5)x+4/5=0$
→$x^2-(19/5)x+\displaystyle\left(\frac{19/5}{2.1}\right)^2+4/5=\displaystyle\left(\frac{19/5}{2.1}\right)^2$
→$x^2-(19/5)x+\displaystyle\left(\frac{19}{10}\right)^2=\displaystyle\left(\frac{19}{10}\right)^2-4/5$
→$(x-19/10)^2=\displaystyle\frac{361}{100}-\frac{4}{5}$→$(x-19/10)^2=\displaystyle\frac{361-80}{100}$
→$x-19/10=\pm\displaystyle\frac{\sqrt{281}}{10}$→$x=\displaystyle\frac{19}{10}\pm\displaystyle\frac{\sqrt{281}}{10}$
→$x_1=\displaystyle\frac{19}{10}+\displaystyle\frac{\sqrt{281}}{10}\bullet$→$x_2=\displaystyle\frac{19}{10}-\displaystyle\frac{\sqrt{281}}{10}\bullet$

h)  $x-\sqrt{x}-6=0$→$(\sqrt{x})^2-\sqrt{x}-6=0$
→$(\sqrt{x}-3)(\sqrt{x}+2)=0$
→$\sqrt{x}-3=0$→$\sqrt{x}=3$→$x_1=9\bullet$
→$\sqrt{x}+2=0$→$\sqrt{x}=-2$→$x_2\not\exists\mathbb{R}\bullet$

i)  $\sqrt{x^2+4}=x+2$→$(\sqrt{x^2+4})^2=(x+2)^2$
→$x^2+4=x^2+4x+4$→$4x=0$→$x=0\bullet$

j)  $x^2-2xy-63y^2=0$→$x^2-2y(x)=63y^2$
→$x^2-2y(x)+(2y/2)^2=63y^2+(2y/2)^2$
→$x^2-2y(x)+(y)^2=63y^2+(y)^2$
→$(x-y)^2=64y^2$→$(x-y)^2-(8y)^2=0$
→$\left[(x-y)+8y\right]\left[(x-y)-8y\right]=0$
→$(x+7y)(x-9y)=0$→$x_1=-7y\bullet$→$x_2=9y\bullet$

k)  $|-(8/5)x-4/3|=2$→$\left|\frac{\displaystyle-24x-20}{\displaystyle15}\right|=2$
→$\frac{\displaystyle-24x-20}{\displaystyle15}=2$→$-24x-20=30$→$x_1=-\displaystyle\frac{25}{12}\bullet$
→$\frac{\displaystyle-24x-20}{\displaystyle15}=-2$→$-24x-20=-30$→$x_2=\displaystyle\frac{5}{12}\bullet$

l)  $6x^2-7x-5=0$→$x^2-(7/6)x=5/6$
→$x^2-(7/6)x+\left(\displaystyle\frac{7/6}{2}\right)^2=5/6+\left(\displaystyle\frac{7/6}{2}\right)^2$
→$x^2-(7/6)x+\left(\displaystyle\frac{7}{12}\right)^2=5/6+\left(\displaystyle\frac{7}{12}\right)^2$
→$(x-7/12)^2=5/6+49/144$
→$x-7/12=\pm\displaystyle\sqrt{\frac{120+49}{144}}$→$x=7/12\pm\displaystyle\sqrt{\frac{169}{144}}$
→$x_1=7/12+13/12$→$x_1=20/12$→$x_1=5/3\bullet$
→$x_2=7/12-13/12$→$x_2=-6/12$→$x_2=-1/2\bullet$

m)  $x^2+3x-1=0$→$x^2+3x=1$
→$x^2+3x+(3/2)^2=1+(3/2)^2$
→$(x+3/2)^2=13/4$→$x+3/2=\pm\sqrt{13/4}$
→$x_1=-3/2+\displaystyle\frac{\sqrt{13}}{2}\bullet$
→$x_2=-3/2-\displaystyle\frac{\sqrt{13}}{2}\bullet$

n)  $x^4-6x^2+5=0$→$a=x^2$→$a^2-6a+5=0$
→$(a-5)(a-1)=0$→$(x^2-5)(x^2-1)=0$→
→$x^2-5=0$→$x=\pm\sqrt{5}$→$x_=\sqrt{5}\bullet$→$x=\pm\sqrt{5}$→$x_=\sqrt{5}\bullet$