Question

Classifique e justifique as equações abaixo como:

l. Possui 1 raiz
ll. Possui 2 raízes
lll. Não possui raiz


a) x²-5x+6=0
b) x²-2x+4=0
c) x²+6x+9=0
d) 2x²+12x+8=0


ME AJUDEM POR FAVOR

Answer 1

Olá

Há uma regra para saber quantas raizes o polinomio tem. São elas:

$\left\{\begin{array}{lll}\Delta\ \textgreater \ 0~ ~\text{(Delta maior que zero)}}~~~~ ~\Longleftarrow \text{Ha duas raizes}\\\Delta\ \textless \ 0~ ~\text{(Delta menor que zero)}}~~~~ ~\Longleftarrow \text{Nao ha raizes}\\\Delta=0~ ~\text{(Delta igual a zero)}}~~~~ ~\Longleftarrow \text{Ha uma raiz}\end{array}\right$

A formula para calcular o $\Delta$ é a seguinte

$\mathsf{\Delta = b^2-4\cdot a\cdot c}$

Sendo 

$\mathsf{'a'~~~~ ~~~ o~ numero~ que~ esta~ junto~ ao~ 'x^2'}\\\\\mathsf{'b'~~~~ ~~~ o~ numero~ que~ esta~ junto~ ao~ x}\\\\\mathsf{'c'~~~~ ~~~ o~ numero~ que~ esta~ sozinho,~ sem~ x}$

A)

$\displaystyle \mathsf{x^2-5x+6=0}\\\\\mathsf{a=1~~~~~~~~b=-5~~~~ ~~~~~c=6}\\\\\mathsf{\Delta=(-5)^2-4\cdot 1\cdot 6}\\\\\mathsf{\Delta=25-24}\\\\\mathsf{\Delta=1} \\\\\Delta~\text{maior que zero. }\\\\\boxed{\text{II. Possui 2 raizes}}$



B)

$\displaystyle \mathsf{x^2-2x+4=0}\\\\\mathsf{a=1~~~~~~~~b=-2~~~~ ~~~~~c=4}\\\\\mathsf{\Delta=(-2)^2-4\cdot 1\cdot 4}\\\\\mathsf{\Delta=4-24}\\\\\mathsf{\Delta=-20} \\\\\Delta~\text{menor que zero. }\\\\\boxed{\text{III. Nao possui raizes}}$


C)

$\displaystyle \mathsf{x^2+6x+9=0}\\\\\mathsf{a=1~~~~~~~~b=6~~~~ ~~~~~c=9}\\\\\mathsf{\Delta=(6)^2-4\cdot 1\cdot 9}\\\\\mathsf{\Delta=36-36}\\\\\mathsf{\Delta=0} \\\\\Delta~\text{igual a zero. }\\\\\boxed{\text{I. Possui 1 raiz}}$



D)

$\displaystyle \mathsf{2x^2+12x+8=0}\\\\\mathsf{a=2~~~~~~~~b=12~~~~ ~~~~~c=8}\\\\\mathsf{\Delta=(12)^2-4\cdot 2\cdot 8}\\\\\mathsf{\Delta=144-64}\\\\\mathsf{\Delta=80} \\\\\Delta~\text{maior que zero. }\\\\\boxed{\text{II. Possui 2 raizes}}$