Resolve as seguintes equações
Monómios e polinómios: Matematicamente Falando 8 - Pág. 148 Ex. 16
Enunciado
Resolve as seguintes equações:
| a) | \(4{x^2} = 81\) |
| b) | \(27 – 3{x^2} = 0\) |
| c) | \({\left( {x – 4} \right)^2} + 9 = 0\) |
| d) | \({\left( {x + 3} \right)^2} – 4 = 0\) |
| e) | \({x^2} – 7 = 5{x^2} – 7\) |
| f) | \(16{x^2} – {\left( {x – 1} \right)^2} = 0\) |
Resolução
Segue a resolução das equações.
| a) | \[\begin{array}{*{20}{l}}{4{x^2} = 81}& \Leftrightarrow &{{x^2} = \frac{{81}}{4}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = – \frac{9}{2}}& \vee &{x = \frac{9}{2}}\end{array}}\end{array}\] | \[S = \left\{ { – \frac{9}{2},\;\frac{9}{2}} \right\}\] |
| b) | \[\begin{array}{*{20}{l}}{27 – 3{x^2} = 0}& \Leftrightarrow &{ – 3{x^2} = – 27}\\{}& \Leftrightarrow &{{x^2} = 9}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = – 3}& \vee &{x = 3}\end{array}}\end{array}\] | \[S = \left\{ { – 3,\;3} \right\}\] |
| c) | \[\begin{array}{*{20}{l}}{{{\left( {x – 4} \right)}^2} + 9 = 0}& \Leftrightarrow &{{{\left( {x – 4} \right)}^2} = – 9}\\{}& \Leftrightarrow &{x \in \emptyset }\\{}&{}&{{\rm{A\;equação\;é\;impossível}}}\end{array}\] | \[S = \left\{ {} \right\}\] |
| d) | \[\begin{array}{*{20}{l}}{{{\left( {x + 3} \right)}^2} – 4 = 0}& \Leftrightarrow &{{{\left( {x + 3} \right)}^2} = 4}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x + 3 = – 2}& \vee &{x + 3 = 2}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = – 5}& \vee &{x = – 1}\end{array}}\end{array}\] | \[S = \left\{ { – 5,\; – 1} \right\}\] |
| e) | \[\begin{array}{*{20}{l}}{{x^2} – 7 = 5{x^2} – 7}& \Leftrightarrow &{{x^2} = 5{x^2}}\\{}& \Leftrightarrow &{4{x^2} = 0}\\{}& \Leftrightarrow &{{x^2} = 0}\\{}& \Leftrightarrow &{x = 0}\end{array}\] | \[S = \left\{ 0 \right\}\] |
| f) | \[\begin{array}{*{20}{l}}{16{x^2} – {{\left( {x – 1} \right)}^2} = 0}& \Leftrightarrow &{{{\left( {4x} \right)}^2} – {{\left( {x – 1} \right)}^2} = 0}\\{}& \Leftrightarrow &{\left( {4x + \left( {x – 1} \right)} \right)\left( {4x – \left( {x – 1} \right)} \right) = 0}\\{}& \Leftrightarrow &{\left( {5x – 1} \right)\left( {3x + 1} \right) = 0}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{5x – 1 = 0}& \vee &{3x + 1 = 0}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{1}{5}}& \vee &{x = – \frac{1}{3}}\end{array}}\end{array}\] | \[S = \left\{ { – \frac{1}{3},\;\frac{1}{5}} \right\}\] |
