To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we first rearrange the equation:
\[ x^2 - 22x + 1 - 15 = 0 \]
This simplifies to:
\[ x^2 - 22x - 14 = 0 \]
Next, we complete the square for the expression \( x^2 - 22x \). To do this, we take half of the coefficient of \( x \) (which is \(-22\)), square it, and add it inside the square:
- Half of \(-22\) is \(-11\).
- Squaring \(-11\) gives \(121\).
We rewrite the equation by adding and subtracting \(121\):
\[ (x^2 - 22x + 121) - 121 - 14 = 0 \]
This simplifies to:
\[ (x - 11)^2 - 135 = 0 \]
Next, we set the equation to zero:
\[ (x - 11)^2 = 135 \]
Taking the square root of both sides, we get:
\[ x - 11 = \sqrt{135} \quad \text{or} \quad x - 11 = -\sqrt{135} \]
Thus,
\[ x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135} \]
To simplify \( \sqrt{135} \):
\[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \]
Now, substituting back into our solutions, we have:
\[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]
So the complete solutions are:
\[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]
Thus, the correct response is:
x equals 11 plus 3√15, 11 minus 3√15
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