Adiya's approach to solving the quadratic equation \(x^2 + 6 = 20x\) by completing the square is not correct based on the steps he outlined.
Here's the correct method for completing the square:
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Rearrange the equation: Start with the given equation: \[ x^2 + 6 - 20x = 0 \] This can be rearranged as: \[ x^2 - 20x + 6 = 0 \]
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Move the constant to the other side: \[ x^2 - 20x = -6 \]
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Complete the square: To complete the square, you need to take the coefficient of \(x\), which is \(-20\), divide it by 2, and then square it. This gives: \[ \left(\frac{-20}{2}\right)^2 = (-10)^2 = 100 \]
Now you add this value (100) to both sides: \[ x^2 - 20x + 100 = -6 + 100 \] This simplifies to: \[ x^2 - 20x + 100 = 94 \]
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Rewrite the left side as a square: The left side can now be factored: \[ (x - 10)^2 = 94 \]
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Solve for \(x\): Taking the square root of both sides: \[ x - 10 = \pm \sqrt{94} \] So: \[ x = 10 \pm \sqrt{94} \]
In summary, Adiya's proposed method of dividing \(6\) by \(2\) and squaring that value is not correct for completing the square in this context. The step involves the coefficient of \(x\) from the linear term, which in this equation is \(-20\). Therefore, Adiya's method for completing the square is incorrect. The correct process involves using the coefficient of the linear term to properly complete the square.
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