Question 157979
I'll do the first two to get you started



# 1




{{{x^2+10x+13}}} Start with the given expression.



Take half of the {{{x}}} coefficient {{{10}}} to get {{{5}}}. In other words, {{{(1/2)(10)=5}}}.



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



{{{x^2+10x+highlight(25-25)+13}}} Now add <font size=4><b>and</b></font> subtract {{{25}}}. Make sure to place this after the "x" term. Notice how {{{25-25=0}}}. So the expression is not changed.



{{{(x^2+10x+25)-25+13}}} Group the first three terms.



{{{(x+5)^2-25+13}}} Factor {{{x^2+10x+25}}} to get {{{(x+5)^2}}}.



{{{(x+5)^2-12}}} Combine like terms.



So after completing the square, {{{x^2+10x+13}}} transforms to {{{(x+5)^2-12}}}. So {{{x^2+10x+13=(x+5)^2-12}}}.



So {{{x^2+10x+13=0}}} is equivalent to {{{(x+5)^2-12=0}}}.



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{{{(x+5)^2-12=0}}} Start with the given equation.



{{{(x+5)^2=0+12}}}Add {{{12}}} to both sides.



{{{(x+5)^2=12}}} Combine like terms.



{{{x+5=0+-sqrt(12)}}} Take the square root of both sides.



{{{x+5=sqrt(12)}}} or {{{x+5=-sqrt(12)}}} Break up the "plus/minus" to form two equations.



{{{x+5=2*sqrt(3)}}} or {{{x+5=-2*sqrt(3)}}}  Simplify the square root.



{{{x=-5+2*sqrt(3)}}} or {{{x=-5-2*sqrt(3)}}} Subtract {{{5}}} from both sides.



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Answer:



So the solutions are {{{x=-5+2*sqrt(3)}}} or {{{x=-5-2*sqrt(3)}}}.




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# 2




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



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



{{{x^2-7x+2=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-7}}}, and {{{c=2}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-7) +- sqrt( (-7)^2-4(1)(2) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-7}}}, and {{{c=2}}}



{{{x = (7 +- sqrt( (-7)^2-4(1)(2) ))/(2(1))}}} Negate {{{-7}}} to get {{{7}}}. 



{{{x = (7 +- sqrt( 49-4(1)(2) ))/(2(1))}}} Square {{{-7}}} to get {{{49}}}. 



{{{x = (7 +- sqrt( 49-8 ))/(2(1))}}} Multiply {{{4(1)(2)}}} to get {{{8}}}



{{{x = (7 +- sqrt( 41 ))/(2(1))}}} Subtract {{{8}}} from {{{49}}} to get {{{41}}}



{{{x = (7 +- sqrt( 41 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (7+sqrt(41))/(2)}}} or {{{x = (7-sqrt(41))/(2)}}} Break up the expression.  



So the answers are {{{x = (7+sqrt(41))/(2)}}} or {{{x = (7-sqrt(41))/(2)}}} 



which approximate to {{{x=6.702}}} or {{{x=0.298}}}