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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In this problem, we need to translate the verbal statement into an algebraic expression to represent the relationship between the negative number and the operations described. Understanding how to form and manipulate these expressions is crucial for solving the equation.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants. In this question, the phrase 'twice the square of the number' indicates that we will be dealing with a quadratic term. Recognizing how to set up and solve quadratic equations is essential for finding the value of the unknown number.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. This process often includes isolating the variable, applying inverse operations, and sometimes factoring or using the quadratic formula. In this problem, we will need to manipulate the equation derived from the problem statement to find the negative number that satisfies the given conditions.