Question

encontre as raízes das equação de segundo grau

a) x elevado a 2 -100 = 0

b) -2x elevado a 2 +50=0

c) -4×elevado a 2 +16=0

d) -3x elevado a 2 +27=0​

Answer 1

Temos casos de equações do 2º grau incompletas com b = 0. Se situam na forma ax² + c = 0, onde os coeficientes são:

• "a" que multiplica x²
• "c" que é o termo independente

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Nestes casos de equações incompletas, a idéia é isolar a incógnita e extrair a raíz quadrada dos membros, assim encontrando dois valores reais para x (se possivel). Veja:

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A )

$\begin{array}{l}\sf x^2-100=0\\\\\sf x^2=100\\\\\sf \sqrt{x^2}=\sqrt{100}\\\\\begin{vmatrix}\sf \!\:x\!\:\end{vmatrix}\sf =10\\\\\therefore~~\sf x=\pm~10\end{array}$

Assim o conjunto solução é:

$\large\begin{array}{l}\sf S=\Big\{-10~~;~~10\Big\}\end{array}$

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B )

$\begin{array}{l}\sf -2x^2+50=0\\\\\sf -2x^2=-50\\\\\sf \Rightarrow~~multiplique~por~-1:\\\\\sf 2x^2=50\\\\\sf x^2=\dfrac{50}{2}\\\\\sf x^2=25\\\\\sf \sqrt{x^2}=\sqrt{25}\\\\\begin{vmatrix}\sf \!\:x\!\:\end{vmatrix}\sf =5\\\\\therefore~~\sf x=\pm~5\end{array}$

Assim o conjunto solução é:

$\large\begin{array}{l}\sf S=\Big\{-5~~;~~5\Big\}\end{array}$

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C )

$\begin{array}{l}\sf -4x^2+16=0\\\\\sf -4x^2=-16\\\\\sf \Rightarrow~~multiplique~por~-1:\\\\\sf 4x^2=16\\\\\sf x^2=\dfrac{16}{4}\\\\\sf x^2=4\\\\\sf \sqrt{x^2}=\sqrt{4}\\\\\begin{vmatrix}\sf\!\: x\!\:\end{vmatrix}\sf =2\\\\\therefore~~\sf x=\pm~2\end{array}$

Assim o conjunto solução é:

$\large\begin{array}{l}\sf S=\Big\{-2~~;~~2\Big\}\end{array}$

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D )

$\begin{array}{l}\sf -3x^2+27=0\\\\\sf -3x^2=-27\\\\\sf \Rightarrow~~multiplique~por~-1:\\\\\sf 3x^2=27\\\\\sf x^2=\dfrac{27}{3}\\\\\sf x^2=9\\\\\sf \sqrt{x^2}=\sqrt{9}\\\\\begin{vmatrix}\sf \!\:x\!\:\end{vmatrix}\sf =3\\\\\therefore~~\sf x=\pm~3\end{array}$

Assim o conjunto solução é:

$\large\begin{array}{l}\sf S=\Big\{-3~~;~~3\Big\}\end{array}$

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Att. Nasgovaskov

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