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Advanced Math Difficulty: Hard

$\sqrt{{(x - 2)}^{2}} = \sqrt{3 x + 34}$

What is the smallest solution to the given equation?

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Explanation

The correct answer is $negative 3$ . Squaring both sides of the given equation yields ${(x - 2)}^{2} = 3 x + 34$ , which can be rewritten as $x^{2} - 4 x + 4 = 3 x + 34$ . Subtracting $3 x$ and $34$ from both sides of this equation yields $x^{2} - 7 x - 30 = 0$ . This quadratic equation can be rewritten as $(x - 10) (x + 3) = 0$ . According to the zero product property, $(x - 10) (x + 3)$ equals zero when either $x - 10 = 0$ or $x + 3 = 0$ . Solving each of these equations for $x$ yields $x equals 10$ or $x = - 3$ . Therefore, the given equation has two solutions, $10$ and $negative 3$ . Of these two solutions, $negative 3$ is the smallest solution to the given equation.