Question 167713

{{{(3x-1)/(4x+7)=1-(6)/(x+7)}}} Start with the given equation.



{{{cross((4x+7))(x+7)((3x-1)/cross((4x+7)))=(4x+7)(x+7)(1)-(4x+7)cross((x+7))((6)/cross((x+7)))}}} Multiply <font size="4"><b>EVERY</b></font> term on both sides by the LCD {{{(4x+7)(x+7)}}}. Doing this will eliminate all of the fractions.




{{{(x+7)(3x-1)=(4x+7)(x+7)-6(4x+7)}}} Multiply and simplify



{{{3x^2+20x-7=4x^2+35x+49-6(4x+7)}}} FOIL



{{{3x^2+20x-7=4x^2+35x+49-24x-42}}} Distribute



{{{3x^2+20x-7=4x^2+11x+7}}} Combine like terms.



{{{3x^2+20x-7-4x^2-11x-7=0}}} Get all terms to the left side.



{{{-x^2+9x-14=0}}} Combine like terms.



{{{-(x^2-9x+14)=0}}} Factor a -1 from the left side



{{{x^2-9x+14=0}}} Divide both sides by -1




Now here are two ways to solve this:



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Method # 1: Factoring



{{{x^2-9x+14=0}}} Start with the given equation


{{{(x-7)(x-2)=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:

{{{x-7=0}}} or  {{{x-2=0}}} 


{{{x=7}}} or  {{{x=2}}}    Now solve for x in each case



So our answers are 


 {{{x=7}}} or  {{{x=2}}} 




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Method # 2 Quadratic Formula



{{{x^2-9x+14=0}}} Start with the given equation.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-9}}}, and {{{c=14}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-9) +- sqrt( (-9)^2-4(1)(14) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-9}}}, and {{{c=14}}}



{{{x = (9 +- sqrt( (-9)^2-4(1)(14) ))/(2(1))}}} Negate {{{-9}}} to get {{{9}}}. 



{{{x = (9 +- sqrt( 81-4(1)(14) ))/(2(1))}}} Square {{{-9}}} to get {{{81}}}. 



{{{x = (9 +- sqrt( 81-56 ))/(2(1))}}} Multiply {{{4(1)(14)}}} to get {{{56}}}



{{{x = (9 +- sqrt( 25 ))/(2(1))}}} Subtract {{{56}}} from {{{81}}} to get {{{25}}}



{{{x = (9 +- sqrt( 25 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (9 +- 5)/(2)}}} Take the square root of {{{25}}} to get {{{5}}}. 



{{{x = (9 + 5)/(2)}}} or {{{x = (9 - 5)/(2)}}} Break up the expression. 



{{{x = (14)/(2)}}} or {{{x =  (4)/(2)}}} Combine like terms. 



{{{x = 7}}} or {{{x = 2}}} Simplify. 



So the answers are {{{x = 7}}} or {{{x = 2}}} 




Either way, you get the same answers.