Question 104367
{{{x^2 = 5x + 2}}} Start with the given equation



{{{x^2-5x=2}}} Subtract 5x from both sides



Take half of the x coefficient -5 to get -2.5 (ie {{{-5/2=-2.5}}})

Now square -2.5 to get 6.25 (ie {{{(-2.5)^2=6.25}}})




{{{x^2-5x+6.25=2+6.25}}} Add this result (6.25) to both sides. Now the expression {{{x^2-5x+6.25}}} is a perfect square trinomial.





{{{(x-2.5)^2=2+6.25}}} Factor {{{x^2-5x+6.25}}} into {{{(x-2.5)^2}}} 




{{{(x-2.5)^2=8.25}}} Combine like terms on the right side


{{{x-2.5=0+-sqrt(8.25)}}} Take the square root of both sides


{{{x=2.5+-sqrt(8.25)}}} Add 2.5 to both sides to isolate x.


So the expression breaks down to

{{{x=2.5+sqrt(8.25)}}} or {{{x=2.5-sqrt(8.25)}}}



So our answer is approximately

{{{x=5.37228132326901}}} or {{{x=-0.372281323269014}}}


Here is visual proof


{{{ graph( 500, 500, -10, 10, -10, 10, x^2-5x-2) }}} graph of {{{y=x^2-5x-2}}}



When we use the root finder feature on a calculator, we would find that the x-intercepts are {{{x=5.37228132326901}}} and {{{x=-0.372281323269014}}}, so this verifies our answer.