Question 151543


{{{9x^2+24x+15=0}}} Start with the given equation.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=9}}}, {{{b=24}}}, and {{{c=15}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(24) +- sqrt( (24)^2-4(9)(15) ))/(2(9))}}} Plug in  {{{a=9}}}, {{{b=24}}}, and {{{c=15}}}



{{{x = (-24 +- sqrt( 576-4(9)(15) ))/(2(9))}}} Square {{{24}}} to get {{{576}}}. 



{{{x = (-24 +- sqrt( 576-540 ))/(2(9))}}} Multiply {{{4(9)(15)}}} to get {{{540}}}



{{{x = (-24 +- sqrt( 36 ))/(2(9))}}} Subtract {{{540}}} from {{{576}}} to get {{{36}}}



{{{x = (-24 +- sqrt( 36 ))/(18)}}} Multiply {{{2}}} and {{{9}}} to get {{{18}}}. 



{{{x = (-24 +- 6)/(18)}}} Take the square root of {{{36}}} to get {{{6}}}. 



{{{x = (-24 + 6)/(18)}}} or {{{x = (-24 - 6)/(18)}}} Break up the expression. 



{{{x = (-18)/(18)}}} or {{{x =  (-30)/(18)}}} Combine like terms. 



{{{x = -1}}} or {{{x = -5/3}}} Simplify. 



So our answers are {{{x = -1}}} or {{{x = -5/3}}} 



--------------------------------------------------


Check:


Let's check the solution {{{x = -1}}}



{{{9x^2+24x+15=0}}} Start with the given equation.



{{{9(-1)^2+24(-1)+15=0}}} Plug in {{{x = -1}}}



{{{9(1)+24(-1)+15=0}}} Square -1 to get 1



{{{9-24+15=0}}} Multiply



{{{0=0}}} Combine like terms. Since the equation is true, this means that {{{x = -1}}} is a solution.


---------------



Let's check the solution {{{x = -5/3}}}



{{{9x^2+24x+15=0}}} Start with the given equation.



{{{9(-5/3)^2+24(-5/3)+15=0}}} Plug in {{{x = -5/3}}}



{{{9(25/9)+24(-5/3)+15=0}}} Square {{{-5/3}}} to get {{{25/9}}}



{{{25+24(-5/3)+15=0}}} Multiply {{{9}}} and {{{25/9}}} to get {{{9(25/9)=cross(9)(25/cross(9))=25}}}. So in this case, the denominator is eliminated.



{{{25-40+15=0}}} Multiply {{{24}}} and {{{-5/3}}} to get {{{24(-5/3)=8*3(-5/3)=8*cross(3)(-5/cross(3))=8(-5)=-40}}}. So in this case, the denominator is eliminated.



{{{0=0}}} Combine like terms. Since the equation is true, this means that {{{x = -5/3}}} is a solution.