To solve the equation \( z^2 + 10z + 6 = 0 \) by completing the square, follow these steps:
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Rearrange the equation into a suitable form:
\[ z^2 + 10z = -6 \]
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Complete the square for the left side. To do this, take half of the coefficient of \( z \) (which is 10), square it, and add it to both sides.
- Half of 10 is 5.
- Squaring 5 gives \( 25 \).
So, we add \( 25 \) to both sides:
\[ z^2 + 10z + 25 = -6 + 25 \] \[ z^2 + 10z + 25 = 19 \]
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Factor the left side:
\[ (z + 5)^2 = 19 \]
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Take the square root of both sides:
\[ z + 5 = \pm \sqrt{19} \]
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Solve for \( z \):
\[ z = -5 \pm \sqrt{19} \]
Thus, the solutions to the equation \( z^2 + 10z + 6 = 0 \) are:
\[ z = -5 + \sqrt{19} \quad \text{and} \quad z = -5 - \sqrt{19} \]
Among the provided options, the correct answer is:
C. \( -5 \pm \sqrt{19} \)