Problem 85 - College Algebra

Question

Evaluate \(x^{2}-x\) for the value of \(x\) satisfying \(4(x-2)+2=4 x-2(2-x)\)

Step-by-Step Solution

Answer

Evaluating the expression \(x^{2}-x\) for the value of \(x\) satisfying \(4(x-2)+2=4x-2(2-x)\) results in \(0\).

Step 1: Simplify both sides of the equation

Expand the equations on both sides to simplify them. On the left side, we have \(4x-8+2\), which simplifies to \(4x-6\). On the right side, expanding yields \(4x - 2(2) + 2x\), which simplifies to \(6x - 4\). Therefore, our equation becomes \(4x - 6 = 6x - 4\).

Step 2: Solve for x

To solve for \(x\), we can subtract \(4x\) from both sides which gives \(2x = 2\). Then divide both sides by 2 to get \(x = 1\).

Step 3: Substitute the value of x into the expression

Now we substitute \(x = 1\) into the given expression \(x^{2}-x\). This gives us \(1^{2} - 1 = 0\).

Key Concepts

Solving Equations

When you have an equation, your main goal is to find out what value or values make it true. This exercise involves balancing one side of an equation with the other. To solve it, start by looking at the equation: \[4(x-2)+2=4x-2(2-x)\]Begin by simplifying both sides. You'll perform operations such as distributing and combining like terms. For instance:

• The left side simplifies to \(4x - 6\).
• The right side simplifies to \(6x - 4\).

Now you have:\[4x - 6 = 6x - 4\]To isolate \(x\), subtract \(4x\) from both sides, which gives the equation: \(2x = 2\). Divide both sides by \(2\) to find \(x = 1\). This process is key for solving equations as it balances both sides while isolating the variable.

Substitution Method

The substitution method is handy once you've solved an equation and found a value for the variable. With \(x = 1\), you can replace \(x\) in the expression you want to evaluate. This expression given in the problem is \(x^2 - x\). To apply the substitution method:

• Replace every instance of \(x\) with \(1\).
• Calculate the new expression: \(1^2 - 1 = 0\).

By substituting \(x\) with its value, we can quickly find the solution to the expression. This step ensures that we're using the right value to get an accurate result, reflecting the solution of the initial equation.

Simplification

Simplification makes an equation or expression easier to work with by reducing it to a more basic form. It's crucial in both solving equations and evaluating expressions. As illustrated in this exercise:To simplify the original equation, you performed these steps:

• Distribute any multiplications across additions or subtractions, such as \(4(x-2)\) becoming \(4x - 8\).
• Combine like terms, such as \(-8 + 2\) becoming \(-6\).

This simplification helps create a clearer path to finding \(x\). Once you know \(x\), simplify the expression \(x^2 - x\) by replacing \(x\) and calculating into \(1^2 - 1\) which gives \(0\). Each simplification step sharpens accuracy, making math challenges easier and more efficient to solve.