To solve the quadratic equation \( x^2 - 16x - 13 = 4 \) using the method of completing the square, we first set the equation in the standard form:
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Rearrange the equation: \[ x^2 - 16x - 13 - 4 = 0 \] \[ x^2 - 16x - 17 = 0 \]
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Move the constant to the other side: \[ x^2 - 16x = 17 \]
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Complete the square:
- Take the coefficient of \( x \), which is \(-16\).
- Divide it by 2: \(-16 \div 2 = -8\).
- Square the result: \((-8)^2 = 64\).
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Add and subtract this square on the left side of the equation: \[ x^2 - 16x + 64 = 17 + 64 \] \[ (x - 8)^2 = 81 \]
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Take the square root of both sides: \[ x - 8 = \pm 9 \]
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Solve for \( x \):
- For the positive case: \[ x - 8 = 9 \implies x = 9 + 8 = 17 \]
- For the negative case: \[ x - 8 = -9 \implies x = -9 + 8 = -1 \]
Thus, the two values that solve the equation \( x^2 - 16x - 13 = 4 \) are:
\[ \boxed{17 \text{ and } -1} \]