Question 189934
{{{-2 = 2x^2 - 5x }}} Start with the given equation.



{{{0 = 2x^2 - 5x +2 }}} Add 2 to both sides.



{{{2x^2 - 5x +2 = 0}}} Rearrange the equation



{{{2(x^2-(5/2)x+1)}}} 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)+1)}}} 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+1)}}} Group the first three terms.



{{{2((x-5/4)^2-25/16+1)}}} Factor {{{x^2-(5/2)x+25/16}}} to get {{{(x-5/4)^2}}}.



{{{2((x-5/4)^2-9/16)}}} Combine like terms.



{{{2(x-5/4)^2+2(-9/16)}}} Distribute.



{{{2(x-5/4)^2-9/8}}} Multiply.



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



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




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{{{2(x-5/4)^2-9/8=0}}} Start with the given equation.



{{{2(x-5/4)^2=0+9/8}}}Add {{{9/8}}} to both sides.



{{{2(x-5/4)^2=9/8}}} Combine like terms.



{{{(x-5/4)^2=(9/8)/(2)}}} Divide both sides by {{{2}}}.



{{{(x-5/4)^2=9/16}}} Reduce.



{{{x-5/4=""+-sqrt(9/16)}}} Take the square root of both sides.



{{{x-5/4=sqrt(9/16)}}} or {{{x-5/4=-sqrt(9/16)}}} Break up the "plus/minus" to form two equations.



{{{x-5/4=3/4}}} or {{{x-5/4=-3/4}}}  Take the square root of {{{9/16}}} to get {{{3/4}}}.



{{{x=5/4+3/4}}} or {{{x=5/4-3/4}}} Add {{{5/4}}} to both sides.



{{{x=2}}} or {{{x=1/2}}} Combine like terms.



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Answer:



So the solutions are {{{x=2}}} or {{{x=1/2}}}.