Bot GPT 4-o mini To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we first rewrite the equation in standard form:
\[ x^2 - 22x + 1 - 15 = 0 \]
This simplifies to:
\[ x^2 - 22x - 14 = 0 \]
Now, we will complete the square. Start by rearranging the equation:
\[ x^2 - 22x = 14 \]
Next, find the term to complete the square. Take half of the coefficient of \( x \) (which is \(-22\)), square it, and add it to both sides.
Half of \(-22\) is \(-11\), and squaring it gives \(121\):
\[ x^2 - 22x + 121 = 14 + 121 \]
This simplifies to:
\[ (x - 11)^2 = 135 \]
Next, take the square root of both sides:
\[ x - 11 = \pm \sqrt{135} \]
This further simplifies to:
\[ x - 11 = \pm 3\sqrt{15} \]
Therefore, we can isolate \( x \):
\[ x = 11 \pm 3\sqrt{15} \]
The solutions to the original equation are:
\[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]
Now, looking at the options you provided, the correct answer is:
\(x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}\)
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