To solve the equation \( x^2 + 35 = 14x \) by completing the square, here are the steps:
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Rearrange the equation: Move all terms to one side of the equation:
\[ x^2 - 14x + 35 = 0 \]
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Complete the square: To do this, focus on the \( x^2 - 14x \) part.
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Take half of the coefficient of \( x \) (which is \(-14\)), square it, and add it to both sides. Half of \(-14\) is \(-7\), and squaring it gives \(49\).
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Therefore, we rewrite the equation as follows:
\[ x^2 - 14x + 49 - 49 + 35 = 0 \]
Simplifying this gives:
\[ (x - 7)^2 - 14 = 0 \]
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Solve for \( x \): Add \( 14 \) to both sides:
\[ (x - 7)^2 = 14 \]
Now, take the square root of both sides:
\[ x - 7 = \pm \sqrt{14} \]
Thus, we can solve for \( x \):
\[ x = 7 \pm \sqrt{14} \]
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Final solution:
The solutions for \( x \) in simplified form are:
\[ x = 7 + \sqrt{14} \quad \text{and} \quad x = 7 - \sqrt{14} \]
These are the values of \( x \) that satisfy the original equation.
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