Question 264629
It turns out that you can't factor {{{2x^2+5x-9}}} over the rationals. So let's follow the instructions and complete the square.




{{{2x^2+5x-9}}} Start with the given expression.



{{{2(x^2+(5/2)x-9/2)}}} Factor out the {{{x^2}}} coefficient {{{2}}}. This step is very important: the {{{x^2}}} coefficient <font size=4><b>must</b></font> be equal to 1.



Take half of the {{{x}}} coefficient {{{5/2}}} to get {{{5/4}}}. In other words, {{{(1/2)(5/2)=5/4}}}.



Now square {{{5/4}}} to get {{{25/16}}}. In other words, {{{(5/4)^2=(5/4)(5/4)=25/16}}}



{{{2(x^2+(5/2)x+highlight(25/16-25/16)-9/2)}}} Now add <font size=4><b>and</b></font> subtract {{{25/16}}} inside the parenthesis. Make sure to place this after the "x" term. Notice how {{{25/16-25/16=0}}}. So the expression is not changed.



{{{2((x^2+(5/2)x+25/16)-25/16-9/2)}}} Group the first three terms.



{{{2((x+5/4)^2-25/16-9/2)}}} Factor {{{x^2+(5/2)x+25/16}}} to get {{{(x+5/4)^2}}}.



{{{2((x+5/4)^2-97/16)}}} Combine like terms.



{{{2(x+5/4)^2+2(-97/16)}}} Distribute.



{{{2(x+5/4)^2-97/8}}} Multiply.



So after completing the square, {{{2x^2+5x-9}}} transforms to {{{2(x+5/4)^2-97/8}}}. So {{{2x^2+5x-9=2(x+5/4)^2-97/8}}}.



So {{{2x^2+5x-9=0}}} is equivalent to {{{2(x+5/4)^2-97/8=0}}}.



Now let's solve {{{2(x+5/4)^2-97/8=0}}}





{{{2(x+5/4)^2-97/8=0}}} Start with the given equation.



{{{2(x+5/4)^2=0+97/8}}}Add {{{97/8}}} to both sides.



{{{2(x+5/4)^2=97/8}}} Combine like terms.



{{{(x+5/4)^2=(97/8)/(2)}}} Divide both sides by {{{2}}}.



{{{(x+5/4)^2=97/16}}} Reduce.



{{{x+5/4=""+-sqrt(97/16)}}} Take the square root of both sides.



{{{x+5/4=sqrt(97/16)}}} or {{{x+5/4=-sqrt(97/16)}}} Break up the "plus/minus" to form two equations.



{{{x+5/4=sqrt(97)/4}}} or {{{x+5/4=-sqrt(97)/4}}}  Simplify the square root.



{{{x=-5/4+sqrt(97)/4}}} or {{{x=-5/4-sqrt(97)/4}}} Subtract {{{5/4}}} from both sides.



{{{x=(-5+sqrt(97))/(4)}}} or {{{x=(-5-sqrt(97))/(4)}}} Combine the fractions.



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Answer:



So the solutions are {{{x=(-5+sqrt(97))/(4)}}} or {{{x=(-5-sqrt(97))/(4)}}}.