Question

Equações irracionais
√9 -x + √4 + x = √25 - x ??

Answer 1

Resolver a equação irracional:

$\mathtt{\sqrt{9-x}+\sqrt{4+x}=\sqrt{25-x}}$


Elevando os dois lados ao quadrado* :

$\mathtt{\big(\sqrt{9-x}+\sqrt{4+x}\big)^{\!2}=\big(\sqrt{25-x}}\big)^{\!2}\\\\ \mathtt{\big(\sqrt{9-x}\big)^{\!2}+2\cdot \sqrt{9-x}\cdot \sqrt{4+x}+ \big(\sqrt{4+x}\big)^{\!2}=\big(\sqrt{25-x}}\big)^{\!2}\\\\ \mathtt{9-\diagup\!\!\!\! x+2\cdot \sqrt{(9-x)(4+x)}+4+\diagup\!\!\!\! x=25-x}\\\\ \mathtt{2\sqrt{(9-x)(4+x)}+13=25-x}\\\\ \mathtt{2\sqrt{(9-x)(4+x)}=25-x-13}\\\\ \mathtt{2\sqrt{(9-x)(4+x)}=12-x}$


Elevando os dois lados ao quadrado* novamente:

$\mathtt{\left[2\sqrt{(9-x)(4+x)}\right]^{\!2}=(12-x)^2}\\\\ \mathtt{4\cdot (9-x)(4+x)=144-24x+x^2}\\\\ \mathtt{4\cdot (36+9x-4x-x^2)=144-24x+x^2}\\\\ \mathtt{4\cdot (36+5x-x^2)=144-24x+x^2}\\\\ \mathtt{144+20x-4x^2=144-24x+x^2}\\\\ \mathtt{0=144-24x+x^2-144-20x+4x^2}$

$\mathtt{4x^2+x^2-24x-20x+144-144=0}\\\\ \mathtt{5x^2-44x=0}\\\\ \mathtt{x(5x-44)=0}\\\\ \begin{array}{rcl} \mathtt{x=0}&~\texttt{ ou }~&\mathtt{5x-44=0}\\\\ \mathtt{x=0}&~\texttt{ ou }~&\mathtt{5x=44}\\\\ \mathtt{x=0}&~\texttt{ ou }~&\mathtt{x=\dfrac{44}{5}} \end{array}$

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*Atenção: Estamos resolvendo uma equação irracional, então devemos testar os valores encontrados para verificar se são soluções da equação dada inicialmente.

•   Testando $\mathtt{x=0}:$

$\mathtt{\sqrt{9-0}+\sqrt{4+0}}\\\\ =\mathtt{\sqrt{9}+\sqrt{4}}\\\\ =\mathtt{3+2}\\\\ =\mathtt{5}\\\\ =\mathtt{\sqrt{25}}\\\\ =\mathtt{\sqrt{25-0}\quad\quad\checkmark}$


•   Testando $\mathtt{x=\dfrac{44}{5}}:$

$\mathtt{\sqrt{9-\dfrac{44}{5}}+\sqrt{4+\dfrac{44}{5}}}\\\\\\ =\mathtt{\sqrt{\dfrac{45}{5}-\dfrac{44}{5}}+\sqrt{\dfrac{20}{5}+\dfrac{44}{5}}}\\\\\\ =\mathtt{\sqrt{\dfrac{1}{5}}+\sqrt{\dfrac{64}{5}}}\\\\\\ =\mathtt{\dfrac{1}{\sqrt{5}}+\dfrac{8}{\sqrt{5}}}\\\\\\ =\mathtt{\dfrac{1+8}{\sqrt{5}}}$

$=\mathtt{\dfrac{9}{\sqrt{5}}}\\\\\\ =\mathtt{\sqrt{\dfrac{81}{5}}}\\\\\\ =\mathtt{\sqrt{\dfrac{125-44}{5}}}\\\\\\ =\mathtt{\sqrt{\dfrac{125}{5}-\dfrac{44}{5}}}\\\\\\ =\mathtt{\sqrt{25-\dfrac{44}{5}}\quad\quad \checkmark}$


Logo, os dois valores encontrados são soluções para a equação irracional dada.


Conjunto solução:   $\mathtt{S=\left\{0,\,\dfrac{44}{5} \right \}}.$


Dúvidas? Comente.


Bons estudos! :-)