Question 191876
I'll do the first three to get you going...



# 1




{{{3x^2-5x-12=0}}} Start with the given equation.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=3}}}, {{{b=-5}}}, and {{{c=-12}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-5) +- sqrt( (-5)^2-4(3)(-12) ))/(2(3))}}} Plug in  {{{a=3}}}, {{{b=-5}}}, and {{{c=-12}}}



{{{x = (5 +- sqrt( (-5)^2-4(3)(-12) ))/(2(3))}}} Negate {{{-5}}} to get {{{5}}}. 



{{{x = (5 +- sqrt( 25-4(3)(-12) ))/(2(3))}}} Square {{{-5}}} to get {{{25}}}. 



{{{x = (5 +- sqrt( 25--144 ))/(2(3))}}} Multiply {{{4(3)(-12)}}} to get {{{-144}}}



{{{x = (5 +- sqrt( 25+144 ))/(2(3))}}} Rewrite {{{sqrt(25--144)}}} as {{{sqrt(25+144)}}}



{{{x = (5 +- sqrt( 169 ))/(2(3))}}} Add {{{25}}} to {{{144}}} to get {{{169}}}



{{{x = (5 +- sqrt( 169 ))/(6)}}} Multiply {{{2}}} and {{{3}}} to get {{{6}}}. 



{{{x = (5 +- 13)/(6)}}} Take the square root of {{{169}}} to get {{{13}}}. 



{{{x = (5 + 13)/(6)}}} or {{{x = (5 - 13)/(6)}}} Break up the expression. 



{{{x = (18)/(6)}}} or {{{x =  (-8)/(6)}}} Combine like terms. 



{{{x = 3}}} or {{{x = -4/3}}} Simplify. 



So the answers are {{{x = 3}}} or {{{x = -4/3}}} 

  


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# 2





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



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



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=1}}}, 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 = (-(1) +- sqrt( (1)^2-4(1)(-2) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=1}}}, and {{{c=-2}}}



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



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



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



{{{x = (-1 +- sqrt( 9 ))/(2(1))}}} Add {{{1}}} to {{{8}}} to get {{{9}}}



{{{x = (-1 +- sqrt( 9 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-1 +- 3)/(2)}}} Take the square root of {{{9}}} to get {{{3}}}. 



{{{x = (-1 + 3)/(2)}}} or {{{x = (-1 - 3)/(2)}}} Break up the expression. 



{{{x = (2)/(2)}}} or {{{x =  (-4)/(2)}}} Combine like terms. 



{{{x = 1}}} or {{{x = -2}}} Simplify. 



So the answers are {{{x = 1}}} or {{{x = -2}}} 

  

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# 3


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



{{{x^2-12-x=0}}} Subtract "x" from both sides.



{{{x^2-x-12=0}}} Rearrange the terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-1}}}, and {{{c=-12}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-1) +- sqrt( (-1)^2-4(1)(-12) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-1}}}, and {{{c=-12}}}



{{{x = (1 +- sqrt( (-1)^2-4(1)(-12) ))/(2(1))}}} Negate {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1-4(1)(-12) ))/(2(1))}}} Square {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1--48 ))/(2(1))}}} Multiply {{{4(1)(-12)}}} to get {{{-48}}}



{{{x = (1 +- sqrt( 1+48 ))/(2(1))}}} Rewrite {{{sqrt(1--48)}}} as {{{sqrt(1+48)}}}



{{{x = (1 +- sqrt( 49 ))/(2(1))}}} Add {{{1}}} to {{{48}}} to get {{{49}}}



{{{x = (1 +- sqrt( 49 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (1 +- 7)/(2)}}} Take the square root of {{{49}}} to get {{{7}}}. 



{{{x = (1 + 7)/(2)}}} or {{{x = (1 - 7)/(2)}}} Break up the expression. 



{{{x = (8)/(2)}}} or {{{x =  (-6)/(2)}}} Combine like terms. 



{{{x = 4}}} or {{{x = -3}}} Simplify. 



So the answers are {{{x = 4}}} or {{{x = -3}}}