Question 90251
{{{x^2 = 5x + 2}}}


{{{x2 -5x =2}}} Subtract 5x from both sides



{{{1(x^2-5x)=2}}} Factor out 1




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

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




{{{1(x^2-5x+6.25)=2}}} Add this result (6.25) inside the parenthesis


{{{1(x^2-5x+6.25)=2+6.25(1)}}} Add 6.25(1) to the other side (remember we factored out a 1)


Now the left side is a complete square


{{{1(x-2.5)^2=2+6.25(1)}}} Factor the left side


{{{1(x-2.5)^2=8.25}}} Multiply and 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


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.