When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

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Define the variable: Let the negative number be represented by \( x \).

Translate the problem into an equation: The sum of 1 and twice the negative number is \( 1 + 2x \). Subtract this from twice the square of the number, which is \( 2x^2 \), and set the result equal to 0. The equation becomes \( 2x^2 - (1 + 2x) = 0 \).

Simplify the equation: Distribute the negative sign to get \( 2x^2 - 1 - 2x = 0 \). Rearrange the terms to form a standard quadratic equation: \( 2x^2 - 2x - 1 = 0 \).

Solve the quadratic equation: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -2 \), and \( c = -1 \). Substitute these values into the formula.

Simplify the quadratic formula: Calculate the discriminant \( b^2 - 4ac \), simplify the square root, and solve for \( x \) to find the two possible solutions for the negative number.

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