Problem 60 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ -6 x(x-1)+x^{2} $$

Step-by-Step Solution

Answer

-5x^2 + 6x

Step 1: Apply Distributive Property

First, use the distributive property of multiplication over subtraction to multiply -6x with each term inside the parenthesis. So, -6x * x = -6x^2 and -6x * -1 = 6x. The expression becomes: \n-6x^2 + 6x + x^2.

Step 2: Combine Like Terms

The like terms in the expression are -6x^2 and x^2, combine these two to simplify the expression. So, the expression becomes: \n-6x^2 + x^2 + 6x = -5x^2 + 6x.

Step 3: Write the final solution

After all the simplification steps, the final solution of the given exercise is -5x^2 + 6x.

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps simplify expressions with parentheses. It allows you to "distribute" a factor across terms inside the parentheses. For example, if you have an expression like \(-6x(x - 1)\), you'll need to apply the distributive property.

This means you multiply \(-6x\) by each term inside the parentheses, one at a time:

• \(-6x \times x = -6x^2\)
• \(-6x \times -1 = 6x\)

After applying the distributive property, the expression becomes \(-6x^2 + 6x + x^2\).

This step is essential for simplifying algebraic expressions because it sets the stage for combining like terms later on.

Combining Like Terms

Combining like terms is another crucial step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In our example, \(-6x^2\) and \(x^2\) are like terms because they both contain \(x^2\).

To combine them, simply add their coefficients together:

• \(-6x^2 + x^2 = -5x^2\)

Now the expression becomes \(-5x^2 + 6x\).

Combining like terms makes expressions more manageable and easier to interpret. It is an important skill in algebra as it helps in reaching the simplest form of an expression.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and arithmetic operations. They might look complex, but breaking them down into smaller, manageable steps makes them much easier to handle.

Consider the expression \(-6x(x-1) + x^2\). It includes the variable \(x\), constants, and operations like multiplication and addition. Initially, it might seem like a puzzle, but using techniques like the distributive property and combining like terms simplifies it successfully to \(-5x^2 + 6x\).

Understanding these expressions requires practice, but once you get the hang of it, you'll find analyzing and simplifying them much less daunting. Remember, each step you take is an inference to the simplest form an expression can have.