Resolve as seguintes equações
Equações de grau superior ao 1.º: Matematicamente Falando 8 - Parte 2 Pág. 81 Ex. 9
Enunciado
Resolve as seguintes equações:
- $x(x-1)=0$
- $(a-1)(a+1)=0$
- ${{x}^{2}}-2x=0$
- ${{a}^{2}}-6a+9=0$
- $4{{y}^{2}}+25=20y$
- ${{c}^{2}}-0,25=0$
- $0,04{{x}^{2}}-0,4x+1=0$
- ${{x}^{2}}=0,01$
Resolução
- Ora,
\[\begin{array}{*{35}{l}}
x(x-1)=0 & \Leftrightarrow & \begin{array}{*{35}{l}}
x=0 & \vee & x-1=0 \\
\end{array} \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
x=0 & \vee & x=1 \\
\end{array} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
(a-1)(a+1)=0 & \Leftrightarrow & \begin{array}{*{35}{l}}
a-1=0 & \vee & a+1=0 \\
\end{array} \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
a=1 & \vee & a=-1 \\
\end{array} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
{{x}^{2}}-2x=0 & \Leftrightarrow & x(x-2)=0 \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
x=0 & \vee & x-2=0 \\
\end{array} \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
x=0 & \vee & x=2 \\
\end{array} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
{{a}^{2}}-6a+9=0 & \Leftrightarrow & {{(a-3)}^{2}}=0 \\
{} & \Leftrightarrow & a-3=0 \\
{} & \Leftrightarrow & a=3 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
4{{y}^{2}}+25=20y & \Leftrightarrow & 4{{y}^{2}}-20y+25=0 \\
{} & \Leftrightarrow & {{(2y-5)}^{2}}=0 \\
{} & \Leftrightarrow & 2y-5=0 \\
{} & \Leftrightarrow & y=\frac{5}{2} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
{{c}^{2}}-0,25=0 & \Leftrightarrow & (c+0,5)(c-0,5)=0 \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
c+0,5=0 & \vee & c-0,5=0 \\
\end{array} \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
c=-0,5 & \vee & c=0,5 \\
\end{array} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
0,04{{x}^{2}}-0,4x+1=0 & \Leftrightarrow & {{(0,2x-1)}^{2}}=0 \\
{} & \Leftrightarrow & 0,2x-1=0 \\
{} & \Leftrightarrow & x=5 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
{{x}^{2}}=0,01 & \Leftrightarrow & {{x}^{2}}-0,01=0 \\
{} & \Leftrightarrow & (x+0,1)(x-0,1)=0 \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
x+0,1=0 & \vee & x-0,1=0 \\
\end{array} \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
x=-0,1 & \vee & x=0,1 \\
\end{array} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
{{y}^{3}}-4y=0 & \Leftrightarrow & y({{y}^{2}}-4)=0 \\
{} & \Leftrightarrow & y(y+2)(y-2)=0 \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
y=0 & \vee & y+2=0 & \vee & y-2=0 \\
\end{array} \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
y=0 & \vee & y=-2 & \vee & y=2 \\
\end{array} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
{{x}^{3}}+4{{x}^{2}}+4x=0 & \Leftrightarrow & x({{x}^{2}}+4x+4)=0 \\
{} & \Leftrightarrow & x{{(x+2)}^{2}}=0 \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
x=0 & \vee & x+2=0 \\
\end{array} \\
{} & \Leftrightarrow & \begin{array}{*{35}{l}}
x=0 & \vee & x=-2 \\
\end{array} \\
\end{array}\]
