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Solve the following quadratic equation using square roots: −9+(x−3)2−4=−4 .(1 point) Responses {2,−8} left brace 2 comma negative 8 right brace {28,−22} left brace 28 comma negative 22 right brace {8,−2} left brace 8 comma negative 2 right brace {16,−10}

1 answer

To solve the quadratic equation \(-9 + (x - 3)^2 - 4 = -4\), let's first simplify it step-by-step:

  1. Start with the given equation: \[ -9 + (x - 3)^2 - 4 = -4 \]

  2. Combine the constant terms \(-9\) and \(-4\): \[ -13 + (x - 3)^2 = -4 \]

  3. To isolate \((x - 3)^2\), add 13 to both sides: \[ (x - 3)^2 = -4 + 13 \] \[ (x - 3)^2 = 9 \]

  4. Now take the square root of both sides, keeping in mind that this can yield two solutions: \[ x - 3 = 3 \quad \text{or} \quad x - 3 = -3 \]

  5. Solving both equations: For \(x - 3 = 3\): \[ x = 6 \]

    For \(x - 3 = -3\): \[ x = 0 \]

Thus, the solutions for the equation are \(x = 6\) and \(x = 0\).

Now, we check the provided response options for the correct one:

  • {2,−8}
  • {28,−22}
  • {8,−2}
  • {16,−10}

None of the options match the solutions we calculated (\(6\) and \(0\)). It seems there might be an error in the options given, or it might involve understanding the problem differently. If the intended operation or original equation is meant to be different, please clarify so I can assist you correctly!