Find all solutions of the equation
|x^2 - 30x - 1| + |x^2 - 30x + 29| = 30.
+1632 Well you definitely want to take out the absolute value signs. And by doing so we have 4 cases:
x^2 - 30x - 1 + x^2 - 30x + 29 = 30
-x^2 + 30x + 1 + x^2 - 30x + 29 = 30
-x^2 + 30x + 1 - x^2 + 30x - 29 = 30
x^2 - 30x - 1 - x^2 + 30x - 29 = 30
Let's solve each one.
1. 2x^2 - 60x + 28 = 30 => 2x^2 - 60x - 2 = 0 => x^2 - 30x - 1 = 0 => \(x = {15\pm\sqrt{226}}\)
2. 30 = 30.
3. -2x^2 + 60x - 58 = 0 => x^2 - 30x + 29 = 0 => (x - 29)(x - 1) = 0 => x = 29, 1
4. - 30 = 30
we have four solutions that are bolded
