Question

Através do discrimiante delta determine a existência e a quantidade de raízes reais em cada equação

A)2x²-3x/2 + 1/4=0​

Answer 1

$\mathrm{Seja\ a\ equa\c{c}\tilde{a}o\ quadr\acute{a}tica}\ 2x^2-\dfrac{3}{2}x+\dfrac{1}{4}=0.$

$\mathrm{Seus\ coeficientes\ s\tilde{a}o}\ a=2,\ b=-\dfrac{3}{2}\ \text{e}\ c=\dfrac{1}{4}.$

$\mathrm{O\ discriminante}\ \Delta\ \mathrm{\acute{e}\ dado\ por}\ \Delta=b^2-4ac.$

$\mathrm{Se}\ \Delta>0,\ \mathrm{existem\ duas\ solu\c{c}\tilde{o}es\ reais\ e\ distintas.}$

$\mathrm{Se}\ \Delta=0,\ \mathrm{existe\ uma\ \acute{u}nica\ solu\c{c}\tilde{a}o\ real.}$

$\mathrm{Se}\ \Delta<0,\ \mathrm{existem\ duas\ solu\c{c}\tilde{o}es\ complexas\ \left(\Im(z)\neq0\right)\ e\ distintas.}$

$\mathrm{Para\ e\ equa\c{c}\tilde{a}o\ do\ problema,\ teremos\ que}\text{:}$

$\Delta=b^2-4ac\Longrightarrow \Delta=\bigg(-\dfrac{3}{2}\bigg)^2-4(2)\bigg(\dfrac{1}{4}\bigg)$

$\Longrightarrow \Delta=\dfrac{9}{4}-\dfrac{8}{4}\ \therefore\ \boxed{\Delta=\dfrac{1}{4}}$

$\mathrm{Como}\ \Delta>0,\ \mathrm{a\ equa\c{c}\tilde{a}o\ possui\ duas\ ra\acute{\i}zes\ reais\ e\ distintas.}$