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Chapter 1: Problem 85

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

Short Answer

Expert verified
The value of the function \(x^{2}-x\) evaluated at the solution of the equation \(4(x-2)+2=4x-2(2-x)\) obtained as \(x=\frac{5}{2}\) is \(\frac{15}{4}\).

Step by step solution

01

Simplify the expression

Start by simplifying both sides of equation \(4(x-2)+2=4x-2(2-x)\). Distribute multiplication to each term inside the parentheses yielding: \(4x-8+2=4x-4x+4\).
02

Simplify further

Simplify the equations further obtaining: \(4x-6=4\). Now, solve for \(x\) by moving terms involving \(x\) to one side and constants to the other side to get \(4x=10\).
03

Solve for x

Divide both sides of the equation by 4 to find the value of \(x\): \(x=\frac{10}{4}\) or \(x=\frac{5}{2}\).
04

Evaluate the function

Substitute \(x=\frac{5}{2}\) into the function \(x^{2}-x\) to find its value at that point. We get \((\frac{5}{2})^{2}-\frac{5}{2}=\frac{25}{4}-\frac{10}{4}=\frac{15}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equations
Equations are like balancing scales. They state that two expressions are equal. Our given equation, 4(x-2)+2 = 4x-2(2-x), is a great example. You'll often encounter equations like these in algebra.

Here, each side of the equation represents a different algebraic expression. To solve it, you need to find the value of the variable that makes the two sides equal. In this case, our variable is \(x\). Understanding how to manipulate and balance equations is crucial for solving them successfully.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a core skill in algebra. It involves reducing the complexity of expressions to make them easier to work with. In our problem, we start by simplifying \(4(x-2)+2\) and \(4x-2(2-x)\).

When simplifying, we distribute the numbers outside the parentheses to each term inside. So, \(4(x-2)\) becomes \(4x-8\), and \(-2(2-x)\) becomes \(-4+2x\).

After distribution, we can combine like terms. So, \(4x-8+2\) turns into \(4x-6\). By simplifying these expressions, we transform a complex equation into something much easier to handle.
Substitution Method
The substitution method is used primarily after finding the value of the variable in an equation. It's a way to check your work or further evaluate expressions linked to the variable.

Once you've solved for \(x\), substitute its value back into another expression to find specific results. In our problem, after obtaining \(x = \frac{5}{2}\), we substitute it into the expression \(x^2 - x\).

By doing this substitution, we can evaluate the expression: \((\frac{5}{2})^2 - \frac{5}{2}\), which once calculated gives us \(\frac{15}{4}\). This shows the power of substitution in verifying solutions and understanding their implications.
Solving for X
Solving for \(x\) involves isolating \(x\) on one side of the equation to determine its value. In algebra, this is a fundamental technique.

In our solution, we find \(x\) by first simplifying the equation to \(4x-6=4\). To isolate \(x\), we adjust the equation to have all terms involving \(x\) on one side, and constants on the other. This gives us \(4x = 10\).

Finally, divide both sides by 4 to solve for \(x\), resulting in \(x = \frac{5}{2}\). This process showcases key algebraic skills, including variable isolation and basic arithmetic operations like division.

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Most popular questions from this chapter

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 square inches. Find the length of a side of the original square.

Solve equation by the method of your choice. $$ \sqrt{3} x^{2}+6 x+7 \sqrt{3}=0 $$

A bank offers two checking account plans. Plan A has a base service charge of 4.00 dollar per month plus 10垄 per check. Plan B charges a base service charge of $2.00 per month plus 15垄 per check. a. Write models for the total monthly costs for each plan if x checks are written. b. Use a graphing utility to graph the models in the same [0, 50, 10] by [0, 10, 1] viewing rectangle. c. Use the graphs (and the intersection feature) to determine for what number of checks per month plan A will be better than plan B. d. Verify the result of part (c) algebraically by solving an inequality.

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.
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