Question

resolver : x^2 - 2x + 5 > 0​

Answer 1

Respuesta:

$\begin{bmatrix}\mathrm{Solucion:}\:&\:\mathrm{Verdadero\:para\:todo\:}x\:\\ \:\mathrm{Notacion\:intervalo}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}$

Explicación paso a paso:

$x^2-2x+5>0$

⇒

$\mathrm{Escribir}\:x^2-2x+5\:\mathrm{en\:la\:forma:\:\:}x^2+2ax+a^2\\\\2a=-2\quad :\quad a=-1\\\\2a=-2\\\\a=-1\\\\\mathrm{Sumar\:y\:restar\:\left(de\:izquierda\:a\:derecha\right)}\:\left(-1\right)^2\\\\x^2-2x+5+\left(-1\right)^2-\left(-1\right)^2\\\\x^2+2ax+a^2=\left(x+a\right)^2\\\\x^2-2x+\left(-1\right)^2=\left(x-1\right)^2\\\\\mathrm{Completar\:el\:cuadrado}\\\\\left(x-1\right)^2+5-\left(-1\right)^2\\\\\mathrm{Simplificar}\\\\\left(x-1\right)^2+4$

⇒

$\left(x-1\right)^2+4>0\\\\\left(x-1\right)^2>-4$