Question 156831

{{{sqrt(3x+4) + x -6=6}}} Start with the given equation.



{{{sqrt(3x+4) + x =12}}} Add 6 to both sides.



{{{sqrt(3x+4)  =-x+12}}} Subtract "x" from both sides.



{{{3x+4  =(-x+12)^2}}} Square both sides to eliminate the square root



{{{3x+4  =x^2-24x+144}}} FOIL



{{{0=x^2-24x+144-3x-4}}} Subtract {{{3x}}} from both sides. Subtract {{{4}}} from both sides.



{{{0=x^2-27x+140}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-27}}}, and {{{c=140}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-27) +- sqrt( (-27)^2-4(1)(140) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-27}}}, and {{{c=140}}}



{{{x = (27 +- sqrt( (-27)^2-4(1)(140) ))/(2(1))}}} Negate {{{-27}}} to get {{{27}}}. 



{{{x = (27 +- sqrt( 729-4(1)(140) ))/(2(1))}}} Square {{{-27}}} to get {{{729}}}. 



{{{x = (27 +- sqrt( 729-560 ))/(2(1))}}} Multiply {{{4(1)(140)}}} to get {{{560}}}



{{{x = (27 +- sqrt( 169 ))/(2(1))}}} Subtract {{{560}}} from {{{729}}} to get {{{169}}}



{{{x = (27 +- sqrt( 169 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (27 +- 13)/(2)}}} Take the square root of {{{169}}} to get {{{13}}}. 



{{{x = (27 + 13)/(2)}}} or {{{x = (27 - 13)/(2)}}} Break up the expression. 



{{{x = (40)/(2)}}} or {{{x =  (14)/(2)}}} Combine like terms. 



{{{x = 20}}} or {{{x = 7}}} Simplify. 



So the possible answers are {{{x = 20}}} or {{{x = 7}}} 

  

However, if you plug {{{x = 20}}} back into the original equation, you'll find that it doesn't work.



So the only solution is {{{x = 7}}}




Now plug in {{{x = 7}}} into AB, BC, and AC



{{{AB=sqrt(3x+4)=sqrt(3(7)+4)=sqrt(21+4)=sqrt(25)=5}}}
{{{BC=x-6=(7)-6=1}}}
{{{AC=6}}}



So the lengths are {{{AB=5}}}, {{{BC=1}}}, and {{{AC=6}}}


Check:


Remember, AB+BC=AC


AB+BC=AC ... Start with the given equation 


5+1=6 ... Plug in {{{AB=5}}}, {{{BC=1}}}, and {{{AC=6}}}


6=6 ... Add Since this equation is true, this verifies the answer.