Question 145389
# 1




{{{4r^2+4r=7}}} Start with the given equation.



{{{4r^2+4r-7=0}}} Get all terms to the left side.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=4}}}, {{{b=4}}}, and {{{c=-7}}}



Let's use the quadratic formula to solve for r



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



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



{{{r = (-4 +- sqrt( 16-4(4)(-7) ))/(2(4))}}} Square {{{4}}} to get {{{16}}}. 



{{{r = (-4 +- sqrt( 16--112 ))/(2(4))}}} Multiply {{{4(4)(-7)}}} to get {{{-112}}}



{{{r = (-4 +- sqrt( 16+112 ))/(2(4))}}} Rewrite {{{sqrt(16--112)}}} as {{{sqrt(16+112)}}}



{{{r = (-4 +- sqrt( 128 ))/(2(4))}}} Add {{{16}}} to {{{112}}} to get {{{128}}}



{{{r = (-4 +- sqrt( 128 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{r = (-4 +- 8*sqrt(2))/(8)}}} 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>)  



{{{r = (-4+8*sqrt(2))/(8)}}} or {{{r = (-4-8*sqrt(2))/(8)}}} Break up the expression.  



{{{r = -1/2+sqrt(2)}}} or {{{r = -1/2-sqrt(2)}}} Reduce


So our answers are {{{r = -1/2+sqrt(2)}}} or {{{r = -1/2-sqrt(2)}}} 



which approximate to {{{r=0.914}}} or {{{r=-1.914}}} 





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# 2





{{{6x^2+53x=6x-35}}} Start with the given equation.



{{{6x^2+53x-6x+35=0}}} Get all terms to the left side.



{{{6x^2+47x+35=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=6}}}, {{{b=47}}}, and {{{c=35}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(47) +- sqrt( (47)^2-4(6)(35) ))/(2(6))}}} Plug in  {{{a=6}}}, {{{b=47}}}, and {{{c=35}}}



{{{x = (-47 +- sqrt( 2209-4(6)(35) ))/(2(6))}}} Square {{{47}}} to get {{{2209}}}. 



{{{x = (-47 +- sqrt( 2209-840 ))/(2(6))}}} Multiply {{{4(6)(35)}}} to get {{{840}}}



{{{x = (-47 +- sqrt( 1369 ))/(2(6))}}} Subtract {{{840}}} from {{{2209}}} to get {{{1369}}}



{{{x = (-47 +- sqrt( 1369 ))/(12)}}} Multiply {{{2}}} and {{{6}}} to get {{{12}}}. 



{{{x = (-47 +- 37)/(12)}}} Take the square root of {{{1369}}} to get {{{37}}}. 



{{{x = (-47 + 37)/(12)}}} or {{{x = (-47 - 37)/(12)}}} Break up the expression. 



{{{x = (-10)/(12)}}} or {{{x =  (-84)/(12)}}} Combine like terms. 



{{{x = -5/6}}} or {{{x = -7}}} Simplify. 



So our answers are {{{x = -5/6}}} or {{{x = -7}}}