if the expression within the absolute value sign is positive, then you get:
x^2 + 6 = 5x
subtract 5x from both sides of that equation and you get:
x^2 - 5x + 6 = 0
factor that quadratic equation to get:
(x-3)(x-2) = 0
solve for x to get:
x = 3
x = 2
if the expression within the absolute value sign is negative, then you get:
x^2 + 6 = -5x
add 5x to both sides of that equation to get:
x^2 + 5x + 6 = 0
factor that quadratic equation to get:
(x+3)*(x+2) = 0
solve for x to get:
x = -3
x = -2
your possible solutions are:
x = 2
x = 3
x = -2
x = -3
x cannot be equal to -2 or -3 because than the absolute value is negative which is impossible because the absolute value of an expression is always positive.
so x has to be 2 or 3 or nothing at all.
replace x with 2 and replace x with 3 and you will see that the original equation is true.
abs(x^2+6) = abs(15) when x = 3 and 5x = 15 when x = 3
abs(x^2+6) = abs(10) when x = 2 and 5x = 10 when x = 2
original equations confirm that the solution is correct.
here's a graph of the two equations.
you can see that they intersect when x = 2 and when x = 3.
