Question 100957
{{{x^2 + 15 = -10x}}} Start with the given equation



{{{x^2 +10x + 15 =0}}} Add 10x to both sides





{{{x^2+10x=-15}}} Subtract 15 from both sides



Take half of the x coefficient 10 to get 5 (ie {{{10/2=5}}})

Now square 5 to get 25 (ie {{{(5)^2=25}}})




{{{x^2+10x+25=-15+25}}} Add this result (25) to both sides. Now the expression {{{x^2+10x+25}}} is a perfect square trinomial.





{{{(x+5)^2=-15+25}}} Factor {{{x^2+10x+25}}} into {{{(x+5)^2}}}  (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>)




{{{(x+5)^2=10}}} Combine like terms on the right side


{{{x+5=0+-sqrt(10)}}} Take the square root of both sides


{{{x=-5+-sqrt(10)}}} Subtract 5 from both sides to isolate x.


So the expression breaks down to

{{{x=-5+sqrt(10)}}} or {{{x=-5-sqrt(10)}}}



So our answer is approximately

{{{x=-1.83772233983162}}} or {{{x=-8.16227766016838}}}


Here is visual proof


{{{ graph( 500, 500, -10, 10, -10, 10, x^2+10x+15) }}} graph of {{{y=x^2+10x+15}}}



When we use the root finder feature on a calculator, we would find that the x-intercepts are {{{x=-1.83772233983162}}} and {{{x=-8.16227766016838}}}, so this verifies our answer.