Evaluate x² - x for the value of x satisfying 4(x - 2) + 2 = 4x - 2(2 - x).

Verified step by step guidance

1. Step 1: Simplify both sides of the equation 4(x - 2) + 2 = 4x - 2(2 - x). Expand the terms using the distributive property: 4(x - 2) becomes 4x - 8, and -2(2 - x) becomes -4 + 2x. Rewrite the equation as 4x - 8 + 2 = 4x - 4 + 2x.
2. Step 2: Combine like terms on both sides of the equation. On the left-hand side, combine -8 and +2 to get -6, so the left-hand side becomes 4x - 6. On the right-hand side, combine -4 and 2x to get 4x + 2x - 4, which simplifies to 6x - 4. The equation now reads: 4x - 6 = 6x - 4.
3. Step 3: Isolate the variable x. Subtract 4x from both sides to eliminate the 4x term on the left-hand side: -6 = 2x - 4. Then, add 4 to both sides to isolate the term with x: -6 + 4 = 2x, which simplifies to -2 = 2x.
4. Step 4: Solve for x by dividing both sides of the equation by 2: x = -2 / 2, which simplifies to x = -1.
5. Step 5: Substitute x = -1 into the expression x^2 - x to evaluate it. Replace x with -1: (-1)^2 - (-1). Simplify the terms: (-1)^2 is 1, and -(-1) is +1. The expression becomes 1 + 1. Simplify further to find the result.

Key Concepts

Solving Linear Equations

To find the value of x in the equation 4(x - 2) + 2 = 4x - 2(2 - x), one must isolate x by simplifying both sides. This involves distributing terms, combining like terms, and rearranging the equation to solve for x. Understanding how to manipulate linear equations is essential for determining the correct value of x.

Substituting Values

Once the value of x is determined, it can be substituted back into the expression x^2 - x. Substitution is a fundamental algebraic technique that allows one to evaluate expressions based on known values. This step is crucial for finding the final result of the expression after solving the equation.

Evaluating Quadratic Expressions

The expression x^2 - x is a quadratic expression, which can be evaluated by plugging in the value of x obtained from the previous steps. Understanding how to compute quadratic expressions involves recognizing the operations of squaring a number and performing basic arithmetic. This concept is vital for arriving at the final numerical answer.