Question 395382
# 1


{{{(5/2)x^2-(3/2)x-1/4=0}}} Start with the given equation.



{{{10x^2-6x-1=0}}} Multiply EVERY term by the LCD 4 to clear out the fractions.



Notice that the quadratic {{{10x^2-6x-1}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=10}}}, {{{B=-6}}}, and {{{C=-1}}}



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(10)(-1) ))/(2(10))}}} Plug in  {{{A=10}}}, {{{B=-6}}}, and {{{C=-1}}}



{{{x = (6 +- sqrt( (-6)^2-4(10)(-1) ))/(2(10))}}} Negate {{{-6}}} to get {{{6}}}. 



{{{x = (6 +- sqrt( 36-4(10)(-1) ))/(2(10))}}} Square {{{-6}}} to get {{{36}}}. 



{{{x = (6 +- sqrt( 36--40 ))/(2(10))}}} Multiply {{{4(10)(-1)}}} to get {{{-40}}}



{{{x = (6 +- sqrt( 36+40 ))/(2(10))}}} Rewrite {{{sqrt(36--40)}}} as {{{sqrt(36+40)}}}



{{{x = (6 +- sqrt( 76 ))/(2(10))}}} Add {{{36}}} to {{{40}}} to get {{{76}}}



{{{x = (6 +- sqrt( 76 ))/(20)}}} Multiply {{{2}}} and {{{10}}} to get {{{20}}}. 



{{{x = (6 +- 2*sqrt(19))/(20)}}} 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*sqrt(19))/(20)}}} or {{{x = (6-2*sqrt(19))/(20)}}} Break up the expression. 



{{{x = (3+sqrt(19))/(10)}}} or {{{x = (3-sqrt(19))/(10)}}}  Reduce. 



So the solutions are {{{x = (3+sqrt(19))/(10)}}} or {{{x = (3-sqrt(19))/(10)}}}



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# 2 




{{{12m^2-192=0}}} Start with the given equation.



Notice that the quadratic {{{12m^2-192}}} is in the form of {{{Am^2+Bm+C}}} where {{{A=12}}}, {{{B=0}}}, and {{{C=-192}}}



Let's use the quadratic formula to solve for "m":



{{{m = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{m = ((0) +- sqrt( (0)^2-4(12)(-192) ))/(2(12))}}} Plug in  {{{A=12}}}, {{{B=0}}}, and {{{C=-192}}}



{{{m = (0 +- sqrt( 0-4(12)(-192) ))/(2(12))}}} Square {{{0}}} to get {{{0}}}. 



{{{m = (0 +- sqrt( 0--9216 ))/(2(12))}}} Multiply {{{4(12)(-192)}}} to get {{{-9216}}}



{{{m = (0 +- sqrt( 0+9216 ))/(2(12))}}} Rewrite {{{sqrt(0--9216)}}} as {{{sqrt(0+9216)}}}



{{{m = (0 +- sqrt( 9216 ))/(2(12))}}} Add {{{0}}} to {{{9216}}} to get {{{9216}}}



{{{m = (0 +- sqrt( 9216 ))/(24)}}} Multiply {{{2}}} and {{{12}}} to get {{{24}}}. 



{{{m = (0 +- 96)/(24)}}} Take the square root of {{{9216}}} to get {{{96}}}. 



{{{m = (0 + 96)/(24)}}} or {{{m = (0 - 96)/(24)}}} Break up the expression. 



{{{m = (96)/(24)}}} or {{{m =  (-96)/(24)}}} Combine like terms. 



{{{m = 4}}} or {{{m = -4}}} Simplify. 



So the solutions are {{{m = 4}}} or {{{m = -4}}} 



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