Question

Quais são as raizes da equação 4X²-4X-5=0?

Answer 1

$4x^2-4x-5=0~~~\Rightarrow~~\left\{\!\begin{array}{l}a=4\\b=-4\\c=-5 \end{array} \right.\\\\\\ \Delta=b^2-4ac\\\\ \Delta=(-4)^2-4\cdot 4\cdot (-5)\\\\ \Delta=16+80\\\\ \Delta=96\\\\ \Delta=2^4\cdot 2\cdot 3\\\\ \Delta=2^4\cdot 6$


$x=\dfrac{-b\pm \sqrt{\Delta}}{2a}\\\\\\ x=\dfrac{-(-4)\pm \sqrt{2^4\cdot 6}}{2\cdot 4}\\\\\\ x=\dfrac{-(-4)\pm \sqrt{2^4}\cdot \sqrt{6}}{2\cdot 4}\\\\\\ x=\dfrac{4\pm 2^2\sqrt{6}}{2\cdot 4}\\\\\\ x=\dfrac{4\pm 4\sqrt{6}}{2\cdot 4}\\\\\\ x=\dfrac{\diagup\!\!\!\! 4\cdot (1\pm \sqrt{6})}{2\cdot \diagup\!\!\!\! 4}\\\\\\ x=\dfrac{1\pm \sqrt{6}}{2}$

$\boxed{\begin{array}{rcl} x=\dfrac{1-\sqrt{6}}{2}&~\text{ ou }~&x=\dfrac{1+\sqrt{6}}{2} \end{array}}$


Conjunto solução: $S=\left\{\dfrac{1-\sqrt{6}}{2}\,,\,\dfrac{1+\sqrt{6}}{2}\right\}$


Bons estudos! :-)