Question

Resolva em R as seguintes equações exponenciais :

A) 11^{2 x^{2} -5x+2} =1

B) 9^{x+1} = \sqrt[3]{3}

Answer 1

a)
$11^{2x^{2}-5x+2}=1\Rightarrow 11^{2x^{2}-5x+2}=11^{0}\Rightarrow 2x^{2}-5x+2=0\\ \\ \\ x= \dfrac{-b\pm \sqrt{b^{2}-4\cdot a\cdot c} }{2\cdot a}\Rightarrow x= \dfrac{5\pm \sqrt{(-5)^{2}-4\cdot 2\cdot 2} }{2\cdot 2}\Rightarrow\\ \\ \\ x= \dfrac{5\pm \sqrt{25-16} }{4}\Rightarrow x= \dfrac{5\pm \sqrt{9} }{4}\Rightarrow x= \dfrac{5\pm 3}{4}\Rightarrow\\ \\ \\ x_{1}= \dfrac{5+3}{4}=2\ \ \ e\ \ x_{2}=\dfrac{5-3}{4}=\dfrac{2}{4}=\dfrac{1}{2}\\ \\ \\ S=\left\{\dfrac{1}{2};2\right \}$


b)
$9^{x+1} = \sqrt[3]{3}\Rightarrow \left(3^{2}\right)^{x+1} =3^{\frac{1}{3}}\Rightarrow 2x+2=\dfrac{1}{3}\Rightarrow 2x=\dfrac{1}{3}-2\Rightarrow\\ \\ \\ 2x=\dfrac{1-6}{3}\Rightarrow 2x=\dfrac{-5}{3}\Rightarrow x=-\dfrac{5}{6}$