To use the quadratic formula to solve the equation \( 10x^2 - 4x + 1 = 6x^2 \), we first need to rearrange the equation into standard form \( ax^2 + bx + c = 0 \).
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Rearranging the equation:
\( 10x^2 - 4x + 1 - 6x^2 = 0 \)
Combine like terms:
\( (10x^2 - 6x^2) - 4x + 1 = 0 \)
This simplifies to:
\( 4x^2 - 4x + 1 = 0 \)
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Identifying coefficients:
Here, we have:
- \( a = 4 \)
- \( b = -4 \)
- \( c = 1 \)
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Using the quadratic formula:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, substitute the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \]
Simplifying this gives:
\[ x = \frac{4 \pm \sqrt{16 - 16}}{8} \]
\[ x = \frac{4 \pm \sqrt{0}}{8} \]
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Simplifying further:
Since \( \sqrt{0} = 0 \), we have:
\[ x = \frac{4 \pm 0}{8} \]
Thus:
\[ x = \frac{4}{8} = \frac{1}{2} \]
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Conclusion:
The solution to the equation \( 10x^2 - 4x + 1 = 6x^2 \) is
\[ x = \frac{1}{2} \]
This is the simplest form of the solution.