Question 448632
I'll do the first two to get you started.


# 1


{{{2x^2+12x-6=0}}} Start with the given equation.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=12}}}, and {{{c=-6}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(12) +- sqrt( (12)^2-4(2)(-6) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=12}}}, and {{{c=-6}}}



{{{x = (-12 +- sqrt( 144-4(2)(-6) ))/(2(2))}}} Square {{{12}}} to get {{{144}}}. 



{{{x = (-12 +- sqrt( 144--48 ))/(2(2))}}} Multiply {{{4(2)(-6)}}} to get {{{-48}}}



{{{x = (-12 +- sqrt( 144+48 ))/(2(2))}}} Rewrite {{{sqrt(144--48)}}} as {{{sqrt(144+48)}}}



{{{x = (-12 +- sqrt( 192 ))/(2(2))}}} Add {{{144}}} to {{{48}}} to get {{{192}}}



{{{x = (-12 +- sqrt( 192 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (-12 +- 8*sqrt(3))/(4)}}} 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 = (-12)/(4) +- (8*sqrt(3))/(4)}}} Break up the fraction.  



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



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



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


========================================================================


# 2

{{{x^2-6x+7=0}}} Start with the given equation.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-6}}}, and {{{c=7}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-6) +- sqrt( (-6)^2-4(1)(7) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-6}}}, and {{{c=7}}}



{{{x = (6 +- sqrt( (-6)^2-4(1)(7) ))/(2(1))}}} Negate {{{-6}}} to get {{{6}}}. 



{{{x = (6 +- sqrt( 36-4(1)(7) ))/(2(1))}}} Square {{{-6}}} to get {{{36}}}. 



{{{x = (6 +- sqrt( 36-28 ))/(2(1))}}} Multiply {{{4(1)(7)}}} to get {{{28}}}



{{{x = (6 +- sqrt( 8 ))/(2(1))}}} Subtract {{{28}}} from {{{36}}} to get {{{8}}}



{{{x = (6 +- sqrt( 8 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



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



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



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



So the answers are {{{x = 3+sqrt(2)}}} or {{{x = 3-sqrt(2)}}}