Resolve cada uma das seguintes equações

Fórmula resolvente da equação do 2.º grau:

$$\begin{array}{*{20}{c}}
{a{x^2} + bx + c = 0}& \Leftrightarrow &{x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}}&{(a \ne 0)}
\end{array}$$

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1. Ora,
   \[\begin{array}{*{20}{l}}{6{x^2} + 5x + 1 = 0}& \Leftrightarrow &{x = \frac{{ – 5 \mp \sqrt {{5^2} – 4 \times 6 \times 1} }}{{2 \times 6}}}\\{}& \Leftrightarrow &{x = \frac{{ – 5 \mp 1}}{{12}}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{{ – 5 – 1}}{{12}}}& \vee &{x = \frac{{ – 5 + 1}}{{12}}}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = – \frac{1}{2}}& \vee &{x = – \frac{1}{3}}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ { – \frac{1}{2}, – \frac{1}{3}} \right\}}\end{array}\]
2. Ora,
   \[\begin{array}{*{20}{l}}{{x^2} – 4x + 4 = 0}& \Leftrightarrow &{x = \frac{{4 \mp \sqrt {{{\left( { – 4} \right)}^2} – 4 \times 1 \times 4} }}{{2 \times 1}}}\\{}& \Leftrightarrow &{x = \frac{{4 \mp 0}}{2}}\\{}& \Leftrightarrow &{x = 2}\\{}&{}&{}\\{}&{}&{S = \left\{ 2 \right\}}\end{array}\]
3. Ora,
   \[\begin{array}{*{20}{l}}{{x^2} – 3x + 2 = 0}& \Leftrightarrow &{x = \frac{{3 \mp \sqrt {{{\left( { – 3} \right)}^2} – 4 \times 1 \times 2} }}{{2 \times 1}}}\\{}& \Leftrightarrow &{x = \frac{{3 \mp 1}}{2}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{{3 – 1}}{2}}& \vee &{x = \frac{{3 + 1}}{2}}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = 1}& \vee &{x = 2}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ {1,\;2} \right\}}\end{array}\]
4. Ora,
   \[\begin{array}{*{20}{l}}{{x^2} – \frac{5}{3}x – \frac{2}{3} = 0}& \Leftrightarrow &{3{x^2} – 5x – 2 = 0}\\{}& \Leftrightarrow &{x = \frac{{5 \mp \sqrt {{{\left( { – 5} \right)}^2} – 4 \times 3 \times \left( { – 2} \right)} }}{{2 \times 3}}}\\{}& \Leftrightarrow &{x = \frac{{5 \mp \sqrt {49} }}{6}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{{5 – 7}}{6}}& \vee &{x = \frac{{5 + 7}}{6}}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = – \frac{1}{3}}& \vee &{x = 2}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ { – \frac{1}{3},\;2} \right\}}\end{array}\]
5. Ora,
   \[\begin{array}{*{20}{l}}{x\left( {x – 8} \right) = – 42 + 5x}& \Leftrightarrow &{{x^2} – 13x + 42 = 0}\\{}& \Leftrightarrow &{x = \frac{{13 \mp \sqrt {{{\left( { – 13} \right)}^2} – 4 \times 1 \times 42} }}{{2 \times 1}}}\\{}& \Leftrightarrow &{x = \frac{{13 \mp 1}}{2}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = 6}& \vee &{x = 7}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ {6,\;7} \right\}}\end{array}\]
6. Ora,
   \[\begin{array}{*{20}{l}}{\frac{x}{4} – \frac{{{{\left( {x – 1} \right)}^2}}}{2} = 0}& \Leftrightarrow &{x – 2\left( {{x^2} – 2x + 1} \right) = 0}\\{}& \Leftrightarrow &{2{x^2} – 5x + 2 = 0}\\{}& \Leftrightarrow &{x = \frac{{5 \mp \sqrt {{{\left( { – 5} \right)}^2} – 4 \times 2 \times 2} }}{{2 \times 2}}}\\{}& \Leftrightarrow &{x = \frac{{5 \mp \sqrt 9 }}{4}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{{5 – 3}}{4}}& \vee &{x = \frac{{5 + 3}}{4}}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{1}{2}}& \vee &{x = 2}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ {\frac{1}{2},\;2} \right\}}\end{array}\]
7. Ora,
   \[\begin{array}{*{20}{l}}{5\left( {3 + x} \right) = \frac{1}{3}{{\left( { – 3 – x} \right)}^2}}& \Leftrightarrow &{45 + 15x = 9 + 6x + {x^2}}\\{}& \Leftrightarrow &{{x^2} – 9x – 36 = 0}\\{}& \Leftrightarrow &{x = \frac{{9 \mp \sqrt {{{\left( { – 9} \right)}^2} – 4 \times 1 \times \left( { – 36} \right)} }}{{2 \times 1}}}\\{}& \Leftrightarrow &{x = \frac{{9 \mp \sqrt {225} }}{2}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{{9 – 15}}{2}}& \vee &{x = \frac{{9 + 15}}{2}}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = – 3}& \vee &{x = 12}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ { – 3,\;12} \right\}}\end{array}\]
8. Ora,
   \[\begin{array}{*{20}{l}}{4x\left( {2x – 5} \right) = 3x – 14}& \Leftrightarrow &{8{x^2} – 23x + 14 = 0}\\{}& \Leftrightarrow &{x = \frac{{23 \mp \sqrt {{{\left( { – 23} \right)}^2} – 4 \times 8 \times 14} }}{{2 \times 8}}}\\{}& \Leftrightarrow &{x = \frac{{23 \mp \sqrt {81} }}{{16}}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{{23 – 9}}{{16}}}& \vee &{x = \frac{{23 + 9}}{{16}}}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = \frac{7}{8}}& \vee &{x = 2}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ {\frac{7}{8},\;2} \right\}}\end{array}\]
9. Ora,
   \[\begin{array}{*{20}{l}}{2{x^2} + 4x – 4 = 0}& \Leftrightarrow &{x = \frac{{ – 4 \mp \sqrt {{4^2} – 4 \times 2 \times \left( { – 4} \right)} }}{{2 \times 2}}}\\{}& \Leftrightarrow &{x = \frac{{ – 4 \mp \sqrt {48} }}{4}}\\{}& \Leftrightarrow &{x = \frac{{ – 4 \mp 4\sqrt 3 }}{4}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x = – 1 – \sqrt 3 }& \vee &{x = 1 + \sqrt 3 }\end{array}}\\{}&{}&{}\\{}&{}&{S = \left\{ {1 – \sqrt 3 ,\;1 + \sqrt 3 } \right\}}\end{array}\]