## anonymous one year ago the sum of the receptacles of 2 consecutive even integers is 9 over 40 this can be represented by the equation shown 1/x+1/x+2=9/40 please use the rational equation to determine all integers...

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1. amilapsn

$\color{blueviolet}{\Large\sf{The~equation ~is ~this,~ok?\\\frac{1}{x}+\frac{1}{x+2}=\frac{9}{40}}}$

2. amilapsn

$\color{red}{\Large\sf{Warning:~I'll~ give~ you~ answer\ here. \ But\ before\ scrolling\ down\\ it \ is\ strongly\ recommended\ to \ think~what~would~ be~the~next~step\\on~your~own.~Agreed?}}$

3. amilapsn

$\color{blueviolet}{\Large\sf{Now~the~first~step:\\All~ you ~have~ to~do ~is~to~get~x~terms~to~the~numerator~of~the\\terms... }}\color{grey}{Give~it~a~try!}$

4. amilapsn

$\color{blueviolet}{\Large\sf{\frac{1}{x}\times x(x+2)+\frac{1}{x+2}\times x(x+2)=\frac{9}{40}\times x(x+2)}}$

5. amilapsn

$\color{blueviolet}{\Large\sf{Now~it's ~a~matter~of~rearranging:\\ }}\color{grey}{\Large\sf{Now ~it's~ your~turn!}}$

6. amilapsn

$\color{blueviolet}{\Large\sf{Have ~you~got~this?\\9x^2-62x-80=0}}$

7. amilapsn

$\color{blue}{\Large\sf{Assuming~you~know~to~solve~quadratic~equations.\\the~roots\ will\ be:}}\\ \color{brown}{\Large\sf{x_1,x_2=\frac{-(-62)\pm\sqrt{(-62)^2-4(9)(-80)}}{2(9)}}}\\\color{blue}{\Large\sf{And \ the\ roots\ will\ be\ your\ answer!!}}$

8. amilapsn

$\color{green}{\Large\sf{There's~an\ easy\ way\ to \ solve\ this \ too....}\\ \Large{Trial~ and~ error\ method....}}$

9. amilapsn

$\color{violet}{\Large\sf{As\ we\ are\ told\ that\ x\ is\ even\ we\\ can\ substitute\ x\ by\ 2y\ where\ y \ is\ an\ integer.\\Then\ the\ equation\ will\ simplify\ to:\\ \frac{1}{y}+\frac{1}{y+1}=\frac{9}{20} }}$

10. amilapsn

$\Large\sf{As\ we\ know,\ there's\ a\ probability\ of:\\20=y(y+1)\\y \ and\ (y+1)\ are\ two\ consecutive\ integers...\\What \ they\ will \ be??}$

11. amilapsn

$\Large\sf{They'll\ be\ 4\ and\ 5.\ Now\ let's\ substitute\ them\ in\ the\\ equation.\ }$