Question 117555
Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{x^2+6*x+4=0}}} ( notice {{{a=1}}}, {{{b=6}}}, and {{{c=4}}})





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




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




{{{x = (-6 +- sqrt( 36+-16 ))/(2*1)}}} Multiply {{{-4*4*1}}} to get {{{-16}}}




{{{x = (-6 +- sqrt( 20 ))/(2*1)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-6 +- 2*sqrt(5))/(2*1)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (-6 +- 2*sqrt(5))/2}}} Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


{{{x = (-6 + 2*sqrt(5))/2}}} or {{{x = (-6 - 2*sqrt(5))/2}}}



Now break up the fraction



{{{x=-6/2+2*sqrt(5)/2}}} or {{{x=-6/2-2*sqrt(5)/2}}}



Simplify



{{{x=-3+sqrt(5)}}} or {{{x=-3-sqrt(5)}}}



So these expressions approximate to


{{{x=-0.76393202250021}}} or {{{x=-5.23606797749979}}}



So our solutions are:

{{{x=-0.76393202250021}}} or {{{x=-5.23606797749979}}}


Notice when we graph {{{x^2+6*x+4}}}, we get:


{{{ graph( 500, 500, -15.2360679774998, 9.23606797749979, -15.2360679774998, 9.23606797749979,1*x^2+6*x+4) }}}


when we use the root finder feature on a calculator, we find that {{{x=-0.76393202250021}}} and {{{x=-5.23606797749979}}}.So this verifies our answer