To solve the equation \( 4x^2 - 2x + 2 - 3x - x^2 = 0 \), we first combine like terms:
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Combine \( 4x^2 \) and \( -x^2 \): \[ 4x^2 - x^2 = 3x^2 \]
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Combine \( -2x \) and \( -3x \): \[ -2x - 3x = -5x \]
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So the equation simplifies to: \[ 3x^2 - 5x + 2 = 0 \]
Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3 \), \( b = -5 \), and \( c = 2 \).
First, calculate the discriminant \( b^2 - 4ac \): \[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 3 \cdot 2 = 24 \] \[ b^2 - 4ac = 25 - 24 = 1 \]
Now, plug the values into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot 3} \] \[ x = \frac{5 \pm 1}{6} \]
This gives us two solutions:
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For \( 5 + 1 \): \[ x = \frac{6}{6} = 1 \]
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For \( 5 - 1 \): \[ x = \frac{4}{6} = \frac{2}{3} \]
Thus, the solutions are \( x = \frac{2}{3} \) and \( x = 1 \).
Putting the smaller x value in the first box, our final answers are: \[ \boxed{\frac{2}{3}} \quad \text{and} \quad \boxed{1} \]