Question 177853
I'll do the first two to get you started


29)




{{{2x^2-6x+1}}} Start with the given expression.



{{{2(x^2-3x+1/2)}}} 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 {{{-3}}} to get {{{-3/2}}}. In other words, {{{(1/2)(-3)=-3/2}}}.



Now square {{{-3/2}}} to get {{{9/4}}}. In other words, {{{(-3/2)^2=(-3/2)(-3/2)=9/4}}}



{{{2(x^2-3x+highlight(9/4-9/4)+1/2)}}} Now add <font size=4><b>and</b></font> subtract {{{9/4}}} inside the parenthesis. Make sure to place this after the "x" term. Notice how {{{9/4-9/4=0}}}. So the expression is not changed.



{{{2((x^2-3x+9/4)-9/4+1/2)}}} Group the first three terms.



{{{2((x-3/2)^2-9/4+1/2)}}} Factor {{{x^2-3x+9/4}}} to get {{{(x-3/2)^2}}}.



{{{2((x-3/2)^2-7/4)}}} Combine like terms.



{{{2(x-3/2)^2+2(-7/4)}}} Distribute.



{{{2(x-3/2)^2-7/2}}} Multiply.



So after completing the square, {{{2x^2-6x+1}}} transforms to {{{2(x-3/2)^2-7/2}}}. So {{{2x^2-6x+1=2(x-3/2)^2-7/2}}}.



So {{{2x^2-6x+1=0}}} is equivalent to {{{2(x-3/2)^2-7/2=0}}}.



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Now let's solve {{{2(x-3/2)^2-7/2=0}}}



{{{2(x-3/2)^2-7/2=0}}} Start with the given equation.



{{{2(x-3/2)^2=0+7/2}}}Add {{{7/2}}} to both sides.



{{{2(x-3/2)^2=7/2}}} Combine like terms.



{{{(x-3/2)^2=(7/2)/(2)}}} Divide both sides by {{{2}}}.



{{{(x-3/2)^2=7/4}}} Reduce.



{{{x-3/2=0+-sqrt(7/4)}}} Take the square root of both sides.



{{{x-3/2=sqrt(7/4)}}} or {{{x-3/2=-sqrt(7/4)}}} Break up the "plus/minus" to form two equations.



{{{x-3/2=sqrt(7)/2}}} or {{{x-3/2=-sqrt(7)/2}}}  Simplify the square root.



{{{x=3/2+sqrt(7)/2}}} or {{{x=3/2-sqrt(7)/2}}} Add {{{3/2}}} to both sides.



{{{x=(3+sqrt(7))/(2)}}} or {{{x=(3-sqrt(7))/(2)}}} Combine the fractions.



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Answer:



So the solutions are {{{x=(3+sqrt(7))/(2)}}} or {{{x=(3-sqrt(7))/(2)}}}.





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30)






{{{-x^2-8x+5}}} Start with the given expression.



{{{-(x^2+8x-5)}}} Factor out the {{{x^2}}} coefficient {{{-1}}}. 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 {{{8}}} to get {{{4}}}. In other words, {{{(1/2)(8)=4}}}.



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



{{{-(x^2+8x+highlight(16-16)-5)}}} Now add <font size=4><b>and</b></font> subtract {{{16}}} inside the parenthesis. Make sure to place this after the "x" term. Notice how {{{16-16=0}}}. So the expression is not changed.



{{{-((x^2+8x+16)-16-5)}}} Group the first three terms.



{{{-((x+4)^2-16-5)}}} Factor {{{x^2+8x+16}}} to get {{{(x+4)^2}}}.



{{{-((x+4)^2-21)}}} Combine like terms.



{{{-(x+4)^2+21}}} Distribute.




So after completing the square, {{{-x^2-8x+5}}} transforms to {{{-(x+4)^2+21}}}. So {{{-x^2-8x+5=-(x+4)^2+21}}}.



So {{{-x^2-8x+5=0}}} is equivalent to {{{-(x+4)^2+21=0}}}.




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Now let's solve {{{-(x+4)^2+21=0}}}



{{{-(x+4)^2+21=0}}} Start with the given equation.



{{{-(x+4)^2=0-21}}}Subtract {{{21}}} from both sides.



{{{-(x+4)^2=-21}}} Combine like terms.



{{{(x+4)^2=(-21)/(-1)}}} Divide both sides by {{{-1}}}.



{{{(x+4)^2=21}}} Reduce.



{{{x+4=0+-sqrt(21)}}} Take the square root of both sides.



{{{x+4=sqrt(21)}}} or {{{x+4=-sqrt(21)}}} Break up the "plus/minus" to form two equations.



{{{x=-4+sqrt(21)}}} or {{{x=-4-sqrt(21)}}} Subtract {{{4}}} from both sides.



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Answer:



So the solutions are {{{x=-4+sqrt(21)}}} or {{{x=-4-sqrt(21)}}}.