Question

resolva a equação exponencial
$8^{x^2} -x= 4^{x-1}$

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{8^{x^2 - x} = 4^{x - 1}}$

$\mathsf{(2^3)^{x^2 - x} = (2^2)^{x - 1}}$

$\mathsf{\not2^{3x^2 - 3x} = \not2^{2x - 2}}$

$\mathsf{3x^2 - 3x = 2x - 2}$

$\mathsf{3x^2 - 5x + 2 = 0}$

$\mathsf{\Delta = b^2 - 4.a.c}$

$\mathsf{\Delta = (-5)^2 - 4.3.2}$

$\mathsf{\Delta = 25 - 24}$

$\mathsf{\Delta = 1}$

$\mathsf{x = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{5 \pm \sqrt{1}}{6} \rightarrow \begin{cases}\mathsf{x' = \dfrac{5 + 1}{6} = \dfrac{6}{6} = 1}\\\\\mathsf{x'' = \dfrac{5 - 1}{6} = \dfrac{4}{6} =\dfrac{2}{3}}\end{cases}}$

$\boxed{\boxed{\mathsf{S = \left\{\dfrac{2}{3};\:1\right\}}}}$

Answer 2

Explicação passo-a-passo:

$8 {}^{x {}^{2} } - x = 4 {}^{x - 1}$

$(2 {}^{3} ) {}^{(x {}^{2}) } - x = (2 {}^{2} ) {}^{x - 1}$

$2 {}^{3x {}^{2} } - x = 2 {}^{2x - 2}$

$3x {}^{2} - x = 2x - 2$

$3x {}^{2} - x - 2x + 2 = 0$

$3x {}^{2} -5x + 2 = 0$

$a = 3$

$b = -5$

$c = 2$

$∆ = b {}^{2} - 4ac$

$∆ = ( - 5) {}^{2} - 4 \times 3 \times 2$

$∆ = 25 - 24$

$∆ = 1$

$x = - \dfrac{b \dfrac{ + }{} \sqrt{∆} }{2a}$

$x = \dfrac{5 \dfrac{ + }{} \sqrt{1} }{2 \times 3}$

$x = \dfrac{5 \dfrac{ + }{} 1}{6}$

$x {}^{1} = \dfrac{5 + 1}{6} = \dfrac{6}{6} = > 1$

$x {}^{2} = \dfrac{5 - 1}{6} = \dfrac{4}{6} = \dfrac{4 {}^{ \div 2} }{6 {}^{ \div 2} } = > \dfrac{2}{3}$