Question 124292
{{{y=2 x^2+10 x+11}}} Start with the given equation



{{{y-11=2 x^2+10 x}}}  Subtract {{{11}}} from both sides



{{{y-11=2(x^2+5x)}}} Factor out the leading coefficient {{{2}}}



Take half of the x coefficient {{{5}}} to get {{{5/2}}} (ie {{{(1/2)(5)=5/2}}}).


Now square {{{5/2}}} to get {{{25/4}}} (ie {{{(5/2)^2=(5/2)(5/2)=25/4}}})





{{{y-11=2(x^2+5x+25/4-25/4)}}} Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of {{{25/4}}} does not change the equation




{{{y-11=2((x+5/2)^2-25/4)}}} Now factor {{{x^2+5x+25/4}}} to get {{{(x+5/2)^2}}}



{{{y-11=2(x+5/2)^2-2(25/4)}}} Distribute



{{{y-11=2(x+5/2)^2-25/2}}} Multiply



{{{y=2(x+5/2)^2-25/2+11}}} Now add {{{11}}} to both sides to isolate y



{{{y=2(x+5/2)^2-3/2}}} Combine like terms




{{{2(x+5/2)^2-3/2=0}}} Now to solve for x, let {{{y=0}}}



{{{2(x+5/2)^2=+3/2}}} Add {{{3/2}}} to both sides



{{{(x+5/2)^2=(3/2)/(2)}}} Divide both sides by {{{2}}}



{{{(x+5/2)^2=3/4}}} Reduce



*[Tex \LARGE x+\frac{5}{2}=\pm \sqrt{\frac{3}{4}}] Take the square root of both sides



*[Tex \LARGE x+\frac{5}{2}=\pm \frac{\sqrt{3}}{2}] Simplify {{{sqrt(3/4)}}} to get {{{sqrt(3)/2}}}



*[Tex \LARGE x=-\frac{5}{2}\pm \frac{\sqrt{3}}{2}]  Subtract {{{5/2}}} from both sides



*[Tex \LARGE x=-\frac{5 \pm \sqrt{3}}{2}]  Combine the fractions




So our solutions are 


*[Tex \LARGE x=-\frac{5 + \sqrt{3}}{2}] or *[Tex \LARGE x=-\frac{5 - \sqrt{3}}{2}]