Question 133088
{{{y=2 x^2-1 x-15}}} Start with the given equation



{{{y+15=2 x^2-1 x}}} Add {{{15}}} to both sides



{{{y+15=2(x^2+(-1/2)x)}}} Factor out the leading coefficient {{{2}}}



Take half of the x coefficient {{{-1/2}}} to get {{{-1/4}}} (ie {{{(1/2)(-1/2)=-1/4}}}).


Now square {{{-1/4}}} to get {{{1/16}}} (ie {{{(-1/4)^2=(-1/4)(-1/4)=1/16}}})





{{{y+15=2(x^2+(-1/2)x+1/16-1/16)}}} Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of {{{1/16}}} does not change the equation




{{{y+15=2((x-1/4)^2-1/16)}}} Now factor {{{x^2+(-1/2)x+1/16}}} to get {{{(x-1/4)^2}}}



{{{y+15=2(x-1/4)^2-2(1/16)}}} Distribute



{{{y+15=2(x-1/4)^2-1/8}}} Multiply



{{{y=2(x-1/4)^2-1/8-15}}} Now add {{{+15}}} to both sides to isolate y



{{{y=2(x-1/4)^2-121/8}}} Combine like terms




{{{2(x-1/4)^2-121/8=0}}} Now to solve for x, let {{{y=0}}}



{{{2(x-1/4)^2=+121/8}}} Add {{{121/8}}} to both sides



{{{(x-1/4)^2=(121/8)/(2)}}} Divide both sides by {{{2}}}



{{{(x-1/4)^2=121/16}}} Reduce



*[Tex \LARGE x+\frac{1}{4}=\pm \sqrt{\frac{121}{16}}] Take the square root of both sides




*[Tex \LARGE x+\frac{1}{4}=\pm \frac{11}{4}] Take the square root of 121 to get 11. Take the square root of 16 to get 4. 



*[Tex \LARGE x=\frac{1}{4}\pm \frac{11}{4}] Add {{{1/4}}} to both sides



Break up the expression


*[Tex \LARGE x=\frac{1}{4}+ \frac{11}{4}] or *[Tex \LARGE x=\frac{1}{4}- \frac{11}{4}]




Combine the fractions


*[Tex \LARGE x=\frac{1+11}{4}] or *[Tex \LARGE x=\frac{1-11}{4}]



Combine like terms



*[Tex \LARGE x=\frac{12}{4}] or *[Tex \LARGE x=-\frac{10}{4}]




Reduce



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





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Answer:

So our solution is 



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