Question

ATIVIDADE 05


EQUAÇÃO INCOMPLETA DO 2° GRAU


Determine, no conjunto R (conjunto dos números reais), o conjunto solução de cada uma das equações do 2º grau.


a) x²-20x=0


b) 2x²-11x=0


c) x² +9x=0


d) -7x² -7x = 0


e) 12x²-24x = 0


f) x²-81=0


g) 16x²-1=0


h) 2x²-18x=0


i) -11x²-44 = 0


j) x² + 100 = 0

Answer 1

O conjunto solução é:

a) $S = \{0,20\}$

b) $S = \{0,\dfrac{11}{2} \}$

c)  $S = \{-9,0\}$

d) $S = \{-1,0\}$

e) $S = \{0,2\}$

f) $S = \{-9,9\}$

g) $S = \{\dfrac{-1}{4} ,\dfrac{1}{4} \}$

h) $S = \{0,9\}$

i) $S= \{ \varnothing \}$

j) $S= \{ \varnothing \}$

Equações do 2º grau são dadas pela generalização:

$a\cdot x^2 +b\cdot x +c= 0 \\\\\ a\neq 0$

Equações incompletas tem b e/ou c =0

Resolvendo:

a)

$x^2 -20 x = 0\\\\x\cdot (x-20)=0\\\\x' =0\\\\x''-20= 0\\\\x'' = 20\\\\\boxed{S= \{0,20\}}$

b)

$2\cdot x^2 -11 \cdot x=0\\\\x\cdot (2\cdot x-11) = 0\\\\x' = 0\\\\2\cdot x''-11=0\\\\2\cdot x''=11\\\\x''=\dfrac{11}{2} \\\\\boxed{S=\{0,\dfrac{11}{2} \}}$

c)

$x^2 +9\cdot x=0\\\\x\cdot(x+9)=0\\\\x'=0\\\\x''+9=0\\\\x''=-9\\\\ \boxed{S= \{-9, 0\}}$

d)

$-7 \cdot x^2 -7\cdot x=0\\\\x\cdot (-7\cdot x -7) =0\\\\x' = 0\\\\-7\cdot x'' -7=0\\\\-7\cdot x'' =7\\\\x''=\dfrac{7}{-7} \\\\x''=-1\\\\\boxed {S = \{-1,0\}}$

e)

$12\cdot x^2 -24\cdot x=0\\\\x\cdot (12\cdot x -24)=0\\\\x'=0\\\\12\cdot x''-24=0\\\\12\cdot x ''=24\\\\x''=\dfrac{24}{12} \\\\x''=2\\\\\boxed{S= \{0,2\}}$

f)

$x^2-81 = 0\\\\x^2=81\\\\x= \sqrt{81}\\\\x=\pm 9 \\\\\boxed{S=\{-9,9\}}$

g)

$16\cdot x^2 -1=0\\\\16\codt x^2=1\\\\x^2 =\dfrac{1}{16} \\\\x=\sqrt{\dfrac{1}{16} } \\\\x= \pm\dfrac{1}{4}\\\\\boxed{S=\{\dfrac{-1}{4}, \dfrac{1}{4}\}}$

h)

$2\cdot x^2 -18\cdot x = 0\\\\x \cdot (2\cdot x -18)=0\\\\x'=0\\\\2\cdot x''-18=0\\\\2 \cdot x''=18\\\\x''=\dfrac{18}{2} \\\\x''=9\\\\\boxed{S=\{0,9\}}$

i)

$-11 \cdot x^2 -44=0\\\\-11\cdot x^2 = 44\\\\x^2 =\frac{44}{-11} \\\\x^2 =-4$

$x \not\in \mathbb{R}$

$\boxed {S=\{S= \varnothing\}}$

j)

$x^2 +100=0\\\\x^2 = -100$

$x \not\in \mathbb{R}$

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