Question

decompor 100 em duas parcelas cujo produto seja 2436 então a menor parcela é

Answer 1

Temos duas parcelas, que juntas, somam 100:

$a+b=100$

E multiplicação delas, resulta em 2.436:

$a.b=2.436$



Isolamos b na primeira equação e trocamos o resultado pelo b da segunda equação:

$b=100-a\\\\ Trocando\ segunda\ equa\c{c}\~ao:\\\\ a(100-a)=2.436\\\\ -a^2+100a=2.436\\\\ -a^2+100a-2.436=0$



Ficamos com uma equação do segundo grau. Aplicando Bháskara para encontrar as raízes:

$Termos:\ \ \ a=-1\ \ \ b=100\ \ \ c=-2.436\\\\ x=\dfrac{-b\pm\sqrt{b^2-4.a.c}}{2.a}\\\\ x=\dfrac{-100\pm\sqrt{(100)^2-4.(-1).(2.436)}}{2.(-1)}\\\\ x=\dfrac{-100\pm\sqrt{10.000-9.744}}{-2}\\\\ x=\dfrac{\not-100\pm\sqrt{256}}{\not-2}\\\\ x=\dfrac{100\pm16}{2}\\\\ x=50\pm8\\\\ x'=58\\\\ x''=42$



$Para\ \underline{a}\ valendo\ 58,\ teremos\ o\ seguinte\ valor\ de\ \underline{b}:\\\\ b=100-58\\\\ b=42\\\\\\\\ Para\ \underline{a}\ valendo\ 42,\ teremos\ o\ seguinte\ valor\ de\ \underline{b}:\\\\ b=100-42\\\\ b=58$


A menor parcela é: 42



Prova real:

$Para\ \underline{a=58}\ e\ \underline{b=42}\\\\ 58+42=100\\\\ 58\times42=2.436\\\\\\ Para\ \underline{a=42}\ e\ \underline{b=58}\\\\ 4+582=100\\\\ 42\times58=2.436$


Bons estudos!

Answer 2

$\left \{ {{x+y=100} \atop {x*y=2436}} \right. \\\\\\x+y=100\to x=-y+100\\\\\\ x*y=2436\to (-y+100)*y=2436\to\\\\ -y^2+100y-2436=0\\\\ a=-1;b=100;c=-2436\\\\ \Delta=b^2-4ac\to \Delta=100^2-4*(-1)*(-2436)\to \Delta=10000-9744\to \\ \Delta=256\\\\\\ y= \frac{-b\pm \sqrt{ \Delta} }{2a} \to y= \frac{-100\pm \sqrt{ 256} }{2*(-1)} \to y= \frac{-100\pm 16 }{-2} \to \\\\\\ y'= \frac{-100+ 16 }{-2} \to y'= \frac{-84 }{-2} \to y'=42\\\\\\ y= \frac{-100-16 }{-2} \to y''= \frac{-116 }{-2} \to y''=58$



$x=-y+100\\\\ x'=-42+100\to x'=58\\\\ x''=-58+100\to x''=42\\\\ \\\\ S= \left \{ (42;58)(58;42)\left \}$


$Verificando:\\\\ \left \{ {{x+y=100} \atop {x*y=2436}} \right.\\\\\\ \left \{ {{42+58=100} \atop {42*58=2436}} \right.$




R.: A menor parcela tem valor igual a  42.