Question 176448
{{{-2d^2-5d+19=0}}} Start with the given equation.



Notice we have a quadratic equation in the form of {{{ad^2+bd+c}}} where {{{a=-2}}}, {{{b=-5}}}, and {{{c=19}}}



Let's use the quadratic formula to solve for d



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



{{{d = (-(-5) +- sqrt( (-5)^2-4(-2)(19) ))/(2(-2))}}} Plug in  {{{a=-2}}}, {{{b=-5}}}, and {{{c=19}}}



{{{d = (5 +- sqrt( (-5)^2-4(-2)(19) ))/(2(-2))}}} Negate {{{-5}}} to get {{{5}}}. 



{{{d = (5 +- sqrt( 25-4(-2)(19) ))/(2(-2))}}} Square {{{-5}}} to get {{{25}}}. 



{{{d = (5 +- sqrt( 25--152 ))/(2(-2))}}} Multiply {{{4(-2)(19)}}} to get {{{-152}}}



{{{d = (5 +- sqrt( 25+152 ))/(2(-2))}}} Rewrite {{{sqrt(25--152)}}} as {{{sqrt(25+152)}}}



{{{d = (5 +- sqrt( 177 ))/(2(-2))}}} Add {{{25}}} to {{{152}}} to get {{{177}}}



{{{d = (5 +- sqrt( 177 ))/(-4)}}} Multiply {{{2}}} and {{{-2}}} to get {{{-4}}}. 



{{{d = (5+sqrt(177))/(-4)}}} or {{{d = (5-sqrt(177))/(-4)}}} Break up the expression.  



{{{d = (-5-sqrt(177))/(4)}}} or {{{d = (-5+sqrt(177))/(4)}}} Reduce



So the answers are {{{d = (-5-sqrt(177))/(4)}}} or {{{d = (-5+sqrt(177))/(4)}}}



which approximate to {{{d=-4.576}}} or {{{d=2.076}}}