Desenvolve e simplifica
Equações de grau superior ao 1.º: Matematicamente Falando 8 - Parte 2 Pág. 81 Ex. 7
Enunciado
Desenvolve e simplifica cada uma das seguintes expressões:
- $15x-{{(x+7)}^{2}}$
- $x(x-1)-{{(x-2)}^{2}}$
- $(x+2)(x-3)+{{(x+1)}^{2}}$
- ${{(x+\frac{1}{2})}^{2}}-{{(x-\frac{1}{2})}^{2}}-\frac{3}{4}(x-1)(x+1)$
Resolução
- Ora,
\[\begin{array}{*{35}{l}}
15x-{{(x+7)}^{2}} & = & 15x-({{x}^{2}}+14x+49) \\
{} & = & 15x-{{x}^{2}}-14x-49 \\
{} & = & -{{x}^{2}}+x-49 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
x(x-1)-{{(x-2)}^{2}} & = & {{x}^{2}}-x-({{x}^{2}}-4x+4) \\
{} & = & {{x}^{2}}-x-{{x}^{2}}+4x-4) \\
{} & = & 3x-4 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
(x+2)(x-3)+{{(x+1)}^{2}} & = & ({{x}^{2}}-3x+2x-6)+({{x}^{2}}+2x+1) \\
{} & = & {{x}^{2}}-3x+2x-6+{{x}^{2}}+2x+1 \\
{} & = & 2{{x}^{2}}+x-5 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
{{(x+\frac{1}{2})}^{2}}-{{(x-\frac{1}{2})}^{2}}-\frac{3}{4}(x-1)(x+1) & = & ({{x}^{2}}+x+\frac{1}{4})-({{x}^{2}}-x+\frac{1}{4})-\frac{3}{4}({{x}^{2}}-1) \\
{} & = & {{x}^{2}}+x+\frac{1}{4}-{{x}^{2}}+x-\frac{1}{4}-\frac{3}{4}{{x}^{2}}+\frac{3}{4} \\
{} & = & -\frac{3}{4}{{x}^{2}}+2x+\frac{3}{4} \\
\end{array}\]Alternativa:
\[\begin{array}{*{35}{l}}
{{(x+\frac{1}{2})}^{2}}-{{(x-\frac{1}{2})}^{2}}-\frac{3}{4}(x-1)(x+1) & = & \left[ (x+\frac{1}{2})+(x-\frac{1}{2}) \right]\times \left[ (x+\frac{1}{2})-(x-\frac{1}{2}) \right]-\frac{3}{4}({{x}^{2}}-1)\,\,\,\,\text{(Porqu }\!\!\hat{\mathrm{e}}\!\!\text{ ?)} \\
{} & = & 2x\times 1-\frac{3}{4}{{x}^{2}}+\frac{3}{4} \\
{} & = & -\frac{3}{4}{{x}^{2}}+2x+\frac{3}{4} \\
\end{array}\]
