Question

Ajutor am nevoie de A si B de la exercitiile E4 si E5.Va rog!Mulțumesc anticipat!

Answer 1

$E4)a) {3}^{x + 1} + {3}^{x} = 108$

${3}^{x} (3 + 1) = 108$

${3}^{x} \times 4 = 108 \: | \div 4$

${3}^{x} = 27 = > x = 3$

$b) {3}^{2x + 1} + {3}^{2x} - {3}^{2x - 1} = 297$

${3}^{2x - 1} ( {3}^{2} + 3 - 1) = 297$

${3}^{2x - 1} (9 + 3 - 1) = 297$

${3}^{2x - 1} \times 11 = 297 \: | \div 11$

${3}^{2x - 1} = 27$

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

$2x - 1 = 3$

$2x = 3 + 1$

$2x = 4 \: | \div 2$

$x = 2$

$E5)a) {4}^{x} + {2}^{x + 1} = 80$

${ ({2}^{x} )}^{2} + {2}^{x} \times 2 = 80$

${2}^{x} = t \: ,t > 0$

${t}^{2} + 2t = 80$

${t}^{2} + 2t - 80 = 0$

$a = 1$

$b = 2$

$c = - 80$

$\Delta = {b}^{2} - 4ac$

$\Delta = {2}^{2} - 4 \times 1 \times ( - 80)$

$\Delta = 4 + 320$

$\Delta = 324$

$t_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}$

$t_{1,2}=\frac{-2\pm\sqrt{324}}{2 \times 1}$

$t_{1,2}=\frac{-2\pm18}{2}$

$t_{1}=\frac{-2 + 18}{2} = \frac{16}{2} = 8$

$t_{2}=\frac{-2 - 18}{2} = - \frac{20}{2} = - 10 \: < 0 = > \: nu \: convine$

${2}^{x} = t = >{2}^{x} = 8 = > x = 3$

$b) {9}^{x} - 10 \times {3}^{x} + 9 = 0$

$({3}^{x} )^{2}- 10 \times {3}^{x} + 9 = 0$

${3}^{x} = t \: ,t > 0$

${t}^{2} - 10t + 9 = 0$

$a = 1$

$b = - 10$

$c = 9$

$\Delta = {b}^{2} - 4ac$

$\Delta = {( - 10)}^{2} - 4 \times 1 \times 9$

$\Delta = 100 - 36$

$\Delta = 64$

$t_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}$

$t_{1,2}=\frac{-( - 10)\pm\sqrt{64}}{2 \times 1}$

$t_{1,2}=\frac{10\pm8}{2}$

$t_{1}=\frac{10 + 8}{2} = \frac{18}{2} = 9$

$t_{2}=\frac{10 - 8}{2} = \frac{2}{2} = 1$

${3}^{x} = t$

$= > {3}^{x} = 9 = > x = 2$

$= > {3}^{x} = 1 = > x = 0$