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# Algebra

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Find all solutions of the equation
|x^2 - 30x - 1| + |x^2 - 30x + 29| = 30.

Dec 20, 2022

#1
+1317
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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

Dec 20, 2022