Question 90222


{{{x^2+6x+2}}} Start with the given equation



{{{x^2+6x=-2}}} Subtract 2 from both sides


{{{1(x^2+6x)=-2}}} Factor out 1




Take half of 6 to get 3 (ie {{{6/2=3}}})

Now square 3 to get 9 (ie {{{3^2=9}}})




{{{1(x^2+6x+9)=-2}}} Add this result (9) inside the parenthesis


{{{1(x^2+6x+9)=-2+9(1)}}} Add 9(1) to the other side (remember we factored out a 1)


Now the left side is a complete square


{{{1(x+3)^2=-2+9(1)}}} Factor the left side


{{{1(x+3)^2=7}}} Multiply and combine like terms on the right side


{{{x+3=0+-sqrt(7)}}} Take the square root of both sides


{{{x=-3+-sqrt(7)}}} Subtract 3 from both sides


So the expression breaks down to

{{{x=-3+sqrt(7)}}} or {{{x=-3-sqrt(7)}}}



So our answer is approximately

{{{x=-0.354248688935409}}} or {{{x=-5.64575131106459}}}


Here is visual proof


{{{ graph( 500, 500, -10, 10, -10, 10, x^2+6x+2) }}} graph of {{{y=x^2+6x+2}}}



When we use the root finder feature on a calculator, we would find that the x-intercepts are {{{x=-0.354248688935409}}} and {{{x=-5.64575131106459}}}, so this verifies our answer.