Question 102791
"24x²-2x=15  solve the equation by factoring"


{{{24x^2-2x=15}}} Start with the given equation


{{{24x^2-2x-15=0}}} Subtract 15 from both sides



{{{(6x-5)(4x+3)=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:


{{{6x-5=0}}} or {{{4x+3=0}}}


{{{x=5/6}}} or {{{x=-3/4}}}  Now solve for x in each case



So our solutions are 


{{{x=5/6}}} or {{{x=-3/4}}} 



Notice if we graph {{{y=24x^2-2x-15}}} we get


{{{ graph(500,500,-10,10,-10,10, 24x^2-2x-15) }}}


and we can see that the graph has roots at {{{x=5/6}}} and {{{x=-3/4}}}, so this verifies our answer.



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"4x²-2x=15  solve the equation by the quadratic formula"



{{{24x^2-2x=15}}} Start with the given equation


{{{24x^2-2x-15=0}}} Subtract 15 from both sides



Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{24*x^2-2*x-15=0}}} ( notice {{{a=24}}}, {{{b=-2}}}, and {{{c=-15}}})





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




{{{x = (2 +- sqrt( (-2)^2-4*24*-15 ))/(2*24)}}} Negate -2 to get 2




{{{x = (2 +- sqrt( 4-4*24*-15 ))/(2*24)}}} Square -2 to get 4  (note: remember when you square -2, you must square the negative as well. This is because {{{(-2)^2=-2*-2=4}}}.)




{{{x = (2 +- sqrt( 4+1440 ))/(2*24)}}} Multiply {{{-4*-15*24}}} to get {{{1440}}}




{{{x = (2 +- sqrt( 1444 ))/(2*24)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (2 +- 38)/(2*24)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (2 +- 38)/48}}} Multiply 2 and 24 to get 48


So now the expression breaks down into two parts


{{{x = (2 + 38)/48}}} or {{{x = (2 - 38)/48}}}


Lets look at the first part:


{{{x=(2 + 38)/48}}}


{{{x=40/48}}} Add the terms in the numerator

{{{x=5/6}}} Divide


So one answer is

{{{x=5/6}}}




Now lets look at the second part:


{{{x=(2 - 38)/48}}}


{{{x=-36/48}}} Subtract the terms in the numerator

{{{x=-3/4}}} Divide


So another answer is

{{{x=-3/4}}}


So our solutions are:

{{{x=5/6}}} or {{{x=-3/4}}}


Notice when we graph {{{24*x^2-2*x-15}}}, we get:


{{{ graph( 500, 500, -13, 15, -13, 15,24*x^2+-2*x+-15) }}}


and we can see that the roots are {{{x=5/6}}} and {{{x=-3/4}}}. This verifies our answer