Question 191078
First, let's complete the square for the left side of the equation {{{4x^2+24x-8=0}}}



{{{4x^2+24x-8}}} Start with the given expression.



{{{4(x^2+6x-2)}}} Factor out the {{{x^2}}} coefficient {{{4}}}. 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 {{{6}}} to get {{{3}}}. In other words, {{{(1/2)(6)=3}}}.



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



{{{4(x^2+6x+highlight(9-9)-2)}}} Now add <font size=4><b>and</b></font> subtract {{{9}}} inside the parenthesis. Make sure to place this after the "x" term. Notice how {{{9-9=0}}}. So the expression is not changed.



{{{4((x^2+6x+9)-9-2)}}} Group the first three terms.



{{{4((x+3)^2-9-2)}}} Factor {{{x^2+6x+9}}} to get {{{(x+3)^2}}}.



{{{4((x+3)^2-11)}}} Combine like terms.



{{{4(x+3)^2+4(-11)}}} Distribute.



{{{4(x+3)^2-44}}} Multiply.



So after completing the square, {{{4x^2+24x-8}}} transforms to {{{4(x+3)^2-44}}}. So {{{4x^2+24x-8=4(x+3)^2-44}}}.



So {{{4x^2+24x-8=0}}} is equivalent to {{{4(x+3)^2-44=0}}}.





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Now let's solve {{{4(x+3)^2-44=0}}}



{{{4(x+3)^2-44=0}}} Start with the given equation.



{{{4(x+3)^2=44}}} Add 44 to both sides



{{{(x+3)^2=(44)/(4)}}} Divide both sides by {{{4}}}.



{{{(x+3)^2=11}}} Reduce.



{{{x+3=""+-sqrt(11)}}} Take the square root of both sides.



{{{x+3=sqrt(11)}}} or {{{x+3=-sqrt(11)}}} Break up the "plus/minus" to form two equations.



{{{x=-3+sqrt(11)}}} or {{{x=-3-sqrt(11)}}} Subtract {{{3}}} from both sides.



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Answer:



So the solutions are {{{x=-3+sqrt(11)}}} or {{{x=-3-sqrt(11)}}}.