Question

Resuelve las ecuaciones

a) x^2 - 6x- 8 = 0
b) x ( -1 + x) = 6
c) 7x^2= 70x
d) 2 ( x + 3) - x ( 2x + 1) = 5
e) 5 ( 2x - 1) + x (x - 1) = 5
f) 12 (x^2 - 1) - 6 ( 2 + x) = 18

Answer 1

$a)\:x^2-6x-8=0$
Para una ecuacion de segundo grado de la forma ax²+bx+c=0 las soluciones son
$x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$x_{1} =\frac{-\left(-6\right)+\sqrt{\left(-6\right)^2-4\cdot \:1\left(-8\right)}}{2\cdot \:1}:\quad 3+\sqrt{17}$

$x_{2} =\frac{-\left(-6\right)-\sqrt{\left(-6\right)^2-4\cdot \:1\left(-8\right)}}{2\cdot \:1}:\quad 3-\sqrt{17}$

$b)\:x\left(-1+x\right)=6$
$-x+x^2=6$
$x^{2} -x-6=0$
Factorizando:
$(x-3)(x+2)=0$
$x_{1} =3$
$x_{2} =-2$

$c)\:7x^2=70x$
$7x^2-70x=0$

$x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$x_{1} =\frac{-\left(-70\right)+\sqrt{\left(-70\right)^2-4\cdot \:7\cdot \:0}}{2\cdot \:7}:\quad 10$

$x_{2} =\frac{-\left(-70\right)-\sqrt{\left(-70\right)^2-4\cdot \:7\cdot \:0}}{2\cdot \:7}:\quad 0$

$d)\:2\left(x+3\right)-x\left(2x+1\right)=5$
$\mathrm{Poner\:los\:parentesis\:utilizando}:\quad \:a\left(b+c\right)=ab+ac$
$2x+6-2x^{2}-x=5$
$-2x^{2}+x+6=5$
$-2x^{2}+x+1=0$
Multiplicar -1 a ambos lados
$2 x^{2} -x-1=0$
Factorizando:
$(x-1)(2x+1)=0$
$x_{1} =1$
$x_{2} =- \frac{1}{2}$

$e)\:5\left(2x-1\right)+x\left(x-1\right)=5$
$\mathrm{Poner\:los\:parentesis\:utilizando}:\quad \:a\left(b+c\right)=ab+ac$
$10x-5+ x^{2} -x=5$
$x^{2} +9x-5=5$
$x^{2} +9x-10=0$
Factorizando:
$(x-1)(x+10)=0$
$x_{1} =1$
$x_{2} =-10$

$f)\:12\left(x^2-1\right)-6\left(2+x\right)=18$
$12 x^{2} -12-12-6x=18$
$12x^2-6x-24=18$
$12x^2-6x-42=0$

$x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$x_{1} =\frac{-\left(-6\right)+\sqrt{\left(-6\right)^2-4\cdot \:12\left(-42\right)}}{2\cdot \:12}:\quad \frac{1+\sqrt{57}}{4}$

$x_{2} =\frac{-\left(-6\right)-\sqrt{\left(-6\right)^2-4\cdot \:12\left(-42\right)}}{2\cdot \:12}:\quad \frac{1-\sqrt{57}}{4}$