\( (x-3)^{2}=1070 \) 2) What are the solutions to the equation? Choose 1 answer: (A) \( x=3 \pm \sqrt{10} \) (B) \( x=-3 \pm \sqrt{10} \) (C) \( x=10 \pm \sqrt{3} \) (D) \( x=-10 \pm \sqrt{3} \)

To solve the equation \( (x - 3)^{2} = 1070 \), begin by taking the square root of both sides: \[ x - 3 = \pm \sqrt{1070} \] Then, isolate \( x \) by adding 3 to both sides: \[ x = 3 \pm \sqrt{1070} \] Now, since \( \sqrt{1070} \) does not perfectly simplify to a whole number, we can approximate it as follows: \[ \sqrt{1070} \approx 32.7 \] So the solutions approximately are: \[ x \approx 3 + 32.7 \quad \text{and} \quad x \approx 3 - 32.7 \] However, none of the provided answers match \( 3 \pm \sqrt{1070} \) specifically; therefore, the right format of the answer is \( x = 3 \pm \sqrt{1070} \). Now, looking at the answer choices provided: (A) \( x=3 \pm \sqrt{10} \) (B) \( x=-3 \pm \sqrt{10} \) (C) \( x=10 \pm \sqrt{3} \) (D) \( x=-10 \pm \sqrt{3} \) None of these options correctly reflect the solutions from our derivation. Thus, we conclude that the solution is \( x = 3 \pm \sqrt{1070} \) — which is not among the given choices. In this case, if you were solving in an exam, you might also consider pointing out potential clerical errors in the options presented. Now, let’s do some problem-solving tips: When faced with equations involving squares, always remember to take the square root with care, applying the positive and negative signs. Often students remember to apply the square root but omit the negative solution, leading to missing potential answers. As for common mistakes, double-check that your radicand isn't simplifying prematurely. It's essential to maintain the integrity of your initial equation and calculate carefully without losing track of essential values. Happy solving!