Question 116587

{{{2x^2+12x=-5}}} Start with the given equation



{{{2x^2+12x+5=0}}} Move all of the terms to the left side



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 {{{2*x^2+12*x+5=0}}} ( notice {{{a=2}}}, {{{b=12}}}, and {{{c=5}}})





{{{x = (-12 +- sqrt( (12)^2-4*2*5 ))/(2*2)}}} Plug in a=2, b=12, and c=5




{{{x = (-12 +- sqrt( 144-4*2*5 ))/(2*2)}}} Square 12 to get 144  




{{{x = (-12 +- sqrt( 144+-40 ))/(2*2)}}} Multiply {{{-4*5*2}}} to get {{{-40}}}




{{{x = (-12 +- sqrt( 104 ))/(2*2)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-12 +- 2*sqrt(26))/(2*2)}}} 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 = (-12 +- 2*sqrt(26))/4}}} Multiply 2 and 2 to get 4


So now the expression breaks down into two parts


{{{x = (-12 + 2*sqrt(26))/4}}} or {{{x = (-12 - 2*sqrt(26))/4}}}



Now break up the fraction



{{{x=-12/4+2*sqrt(26)/4}}} or {{{x=-12/4-2*sqrt(26)/4}}}



Simplify



{{{x=-3+sqrt(26)/2}}} or {{{x=-3-sqrt(26)/2}}}



So these expressions approximate to


{{{x=-0.450490243203608}}} or {{{x=-5.54950975679639}}}



So our solutions are:

{{{x=-0.450490243203608}}} or {{{x=-5.54950975679639}}}


Notice when we graph {{{2*x^2+12*x+5}}}, we get:


{{{ graph( 500, 500, -15.5495097567964, 9.54950975679639, -15.5495097567964, 9.54950975679639,2*x^2+12*x+5) }}}


when we use the root finder feature on a calculator, we find that {{{x=-0.450490243203608}}} and {{{x=-5.54950975679639}}}.So this verifies our answer