Question 389186
Solving by use of the quadratic formula:



{{{x^2+8x-2=0}}} Start with the given equation.



Notice that the quadratic {{{x^2+8x-2}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=1}}}, {{{B=8}}}, 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 = (-(8) +- sqrt( (8)^2-4(1)(-2) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=8}}}, and {{{C=-2}}}



{{{x = (-8 +- sqrt( 64-4(1)(-2) ))/(2(1))}}} Square {{{8}}} to get {{{64}}}. 



{{{x = (-8 +- sqrt( 64--8 ))/(2(1))}}} Multiply {{{4(1)(-2)}}} to get {{{-8}}}



{{{x = (-8 +- sqrt( 64+8 ))/(2(1))}}} Rewrite {{{sqrt(64--8)}}} as {{{sqrt(64+8)}}}



{{{x = (-8 +- sqrt( 72 ))/(2(1))}}} Add {{{64}}} to {{{8}}} to get {{{72}}}



{{{x = (-8 +- sqrt( 72 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-8 +- 6*sqrt(2))/(2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (-8)/(2) +- (6*sqrt(2))/(2)}}} Break up the fraction.  



{{{x = -4 +- 3*sqrt(2)}}} Reduce.  



{{{x = -4+3*sqrt(2)}}} or {{{x = -4-3*sqrt(2)}}} Break up the expression.  



So the solutions are {{{x = -4+3*sqrt(2)}}} or {{{x = -4-3*sqrt(2)}}} 



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Solving by use of completing the square:



First we need to complete the square for the expression {{{x^2+8x-2}}}



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



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



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



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



{{{(x+4)^2-18}}} Combine like terms.



So after completing the square, {{{x^2+8x-2}}} transforms to {{{(x+4)^2-18}}}. So {{{x^2+8x-2=(x+4)^2-18}}}.



So {{{x^2+8x-2=0}}} is equivalent to {{{(x+4)^2-18=0}}}.



Now let's solve {{{(x+4)^2-18=0}}}



{{{(x+4)^2-18=0}}} Start with the given equation.



{{{(x+4)^2=0+18}}}Add {{{18}}} to both sides.



{{{(x+4)^2=18}}} Combine like terms.



{{{x+4=""+-sqrt(18)}}} Take the square root of both sides.



{{{x+4=sqrt(18)}}} or {{{x+4=-sqrt(18)}}} Break up the "plus/minus" to form two equations.



{{{x+4=3*sqrt(2)}}} or {{{x+4=-3*sqrt(2)}}}  Simplify the square root.



{{{x=-4+3*sqrt(2)}}} or {{{x=-4-3*sqrt(2)}}} Subtract {{{4}}} from both sides.



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Answer:



So the solutions are {{{x=-4+3*sqrt(2)}}} or {{{x=-4-3*sqrt(2)}}}.



Notice how we get the same answers. So either method works.



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