Question 186422
"two numbers when added together equals -6 " translates to {{{x+y=-6}}} and


"same two numbers when multiplied equals -12" translates to {{{x*y=-12}}}



{{{x+y=-6}}} Start with the first equation



{{{y=-6-x}}} Subtract "x" from both sides.



{{{y=-x-6}}} Rearrange the terms.



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{{{x*y=-12}}} Move onto the second equation



{{{x*(-x-6)=-12}}} Plug in {{{y=-x-6}}}



{{{x*(-x)-x(6)=-12}}} Distribute



{{{-x^2-6x=-12}}} Multiply



{{{x^2+6x=12}}} Multiply EVERY term by -1 to make the leading coefficient positive.



{{{x^2+6x-12=0}}} Subtract 12 from both sides.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=6}}}, and {{{c=-12}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(6) +- sqrt( (6)^2-4(1)(-12) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=6}}}, and {{{c=-12}}}



{{{x = (-6 +- sqrt( 36-4(1)(-12) ))/(2(1))}}} Square {{{6}}} to get {{{36}}}. 



{{{x = (-6 +- sqrt( 36--48 ))/(2(1))}}} Multiply {{{4(1)(-12)}}} to get {{{-48}}}



{{{x = (-6 +- sqrt( 36+48 ))/(2(1))}}} Rewrite {{{sqrt(36--48)}}} as {{{sqrt(36+48)}}}



{{{x = (-6 +- sqrt( 84 ))/(2(1))}}} Add {{{36}}} to {{{48}}} to get {{{84}}}



{{{x = (-6 +- sqrt( 84 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-6 +- 2*sqrt(21))/(2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (-6)/(2) +- (2*sqrt(21))/(2)}}} Break up the fraction.  



{{{x = -3 +- sqrt(21)}}} Reduce.  



{{{x = -3+sqrt(21)}}} or {{{x = -3-sqrt(21)}}} Break up the expression.  



So the answers are {{{x = -3+sqrt(21)}}} or {{{x = -3-sqrt(21)}}} 



which approximate to {{{x=1.583}}} or {{{x=-7.583}}} (use a calculator to find the approximate values).



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Answer:



So the numbers are roughly 1.583 and -7.583 (in either order)



Note:  1.583 + -7.583 = -6 and  1.583 * -7.583 = -12.003889 (which is very close)