Problem 18 - Algebra 1

Question

Use the distributive property and mental math to simplify the expression. $$ -6 w-12-3 w+2 x^{2} $$

Step-by-Step Solution

Answer

The simplified version of the expression \(-6w -12 -3w + 2x^2\) is \(2x^2 -9w -12\).

Step 1: Group-like Terms Together

In the expression \(-6w -12 -3w + 2x^2\), we can first group together the similar types of terms together: \(-6w -3w + -12 + 2x^2\). Here we moved the terms that had 'w' together, and clubbed the constant (-12) and the term with 'x' together.

Step 2: Apply Distributive Property

After grouping, apply the distributive property to simplify the 'w' terms: \((-6 -3)w + (-12) + 2x^2\), which simplifies further to \(-9w - 12 + 2x^2\). The distributive property allows us to combine like terms by adding or subtracting their coefficients.

Step 3: Rearrange for Better Reading

Lastly, to better structure the simplified expression, you might want to place the term with the highest exponent first: \(2x^2 -9w -12\). Higher degree terms are usually placed first for better readability.

Key Concepts

Understanding Like Terms

In algebra, identifying and grouping "like terms" is a fundamental skill. Like terms refer to terms in an expression that have the same variable raised to the same power. For example, in the expression \(-6w - 12 - 3w + 2x^2\), \(-6w\) and \(-3w\) are like terms because both have the variable \(w\). Similarly, constant terms (numbers without variables) like \(-12\) are considered like terms.

• To easily identify like terms, focus on their variables and exponents.
• Combine like terms by simply adding or subtracting their coefficients.

This process not only simplifies an expression but also makes it easier to follow through with other algebraic manipulations like applying the distributive property. Grouping like terms is the first step towards simplifying algebraic expressions.

Simplifying Expressions with the Distributive Property

When you work with algebraic expressions, simplification can help clarify complex equations. The distributive property is a powerful tool in this process. Essentially, it allows you to "distribute" a common factor across terms within parentheses. It's often used to simplify the adding or subtracting of like terms.Using the given problem as an example, the distributive property helped us tackle the \(-6w - 12 - 3w + 2x^2\) expression by combining the like \(w\) terms:

• By applying the distributive property, expressions like \((-6 - 3)w\) simplify to \(-9w\).
• Once like terms are combined, you gain a clearer, more readable expression.

Not only does this make the calculation easier, but it also leads to better understanding and presentation of algebraic expressions. Hence, simplifying with the distributive property is indispensable in algebra.

Using Mental Math to Simplify Expressions

Mental math techniques can often simplify the process of algebraic calculations, making it quicker and less error-prone. When you encounter expressions like the example \(-6w - 12 - 3w + 2x^2\), mental math enables you to swiftly combine like terms.Here are some tips to improve your mental math skills:

• Recognize patterns in numbers, such as adding or subtracting coefficients.
• Visualize combining numbers: grouping \(-6 - 3\) mentally to immediately see \(-9\).
• Practice frequently with simple expressions to boost confidence and speed.

Using mental math allows you to solve mathematical problems without writing every step down, making the process of simplifying algebraic expressions more intuitive and efficient. Once your mental math skills are honed, you'll find it easier to follow through on algebra problems and check your work efficiently.