Question

1) Resolva as seguintes equações incompletas do 2° grau

a) x² - 49x = 0
b) x² + 3x = 0
c) 2x² - 50 = 0
d) 7x² - 7 = 0
e) 5x² - 15 = 0

Answer 1

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Equações incompletas de 2º grau

$\boxed{\begin{array}{l}\sf S\tilde ao~equac_{\!\!,}\tilde oes~que~assumem~as~formas\\\boxed{\tt ax^2+bx=0}~\sf ou~\boxed{\tt ax^2+c=0},com~a,b\in\mathbb{R},a\ne0\end{array}}$

Propriedade do produto nulo

$\boxed{\begin{array}{l}\sf se~a\cdot b=0\implies a=0~ou~b=0\end{array}}$

Propriedade dos números reais

$\boxed{\begin{array}{l}\sf Sendo~a~e~b~dois~n\acute umeros~reais~quaisquer\\\sf e~a^2=b,ent\tilde ao~a=+\sqrt{b}~ou~a=-\sqrt{b} \end{array}}$

$\tt a)~\sf x^2-49x=0\\\sf x\cdot(x-49)=0\\\sf x=0~ou~x-49=0\implies x=49\\\huge\boxed{\boxed{\boxed{\boxed{\sf S=\{0,49\}}}}}\blue{\checkmark}\\\tt b)~\sf x^2+3x=0\\\sf x\cdot(x+3)=0\\\sf x=0~ou~x+3=0\implies x=-3\\\huge\boxed{\boxed{\boxed{\boxed{\sf S=\{-3,0\}}}}}\blue{\checkmark}\\\tt c)~\sf2x^2-50=0\\\sf 2x^2=50\\\sf x^2=\dfrac{50}{2}\\\sf x^2=25\\\sf x=\pm\sqrt{25}\\\sf x=\pm5\\\huge\boxed{\boxed{\boxed{\boxed{\sf S=\{-5,5\}}}}}\blue{\checkmark}$

$\tt d)~\sf 7x^2-7=0\\\sf 7x^2=7\\\sf x^2=\dfrac{7}{7}\\\sf x^2=1\\\sf x=\pm\sqrt{1}\\\sf x=\pm1\\\huge\boxed{\boxed{\boxed{\boxed{\sf S=\{-1,1\}}}}}\blue{\checkmark}\\\tt e)~\sf 5x^2-15=0\\\sf 5x^2=15\\\sf x^2=\dfrac{15}{5}\\\sf x^2=3\\\sf x=\pm\sqrt{3}\\\huge\boxed{\boxed{\boxed{\boxed{\sf S=\{-\sqrt{3},\sqrt{3}\}}}}}\blue{\checkmark}$