Problem 95 - College Algebra

Question

Simplify algebraic expression. \(18 x^{2}+4-\left[6\left(x^{2}-2\right)+5\right]\)

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \(12x^{2} + 11\).

Step 1: Distribute Inside Brackets

Distribute the 6 inside the brackets \(6*(x^{2}-2)\) to give \(6*x^{2}-12\). Now the algebraic expression becomes \(18x^{2} + 4 - (6x^{2} - 12 + 5)\).

Step 2: Simplify the Inside Expression

Inside the brackets calculate \(6x^{2} - 12 + 5\) to give \(6x^{2} - 7\). Now the algebraic expression becomes \(18x^{2} + 4 - (6x^{2} - 7)\).

Step 3: Distribute the Negative

Distribute the negative sign to both terms inside the brackets to get -\(6x^{2}\) and 7. The algebraic expression then becomes \(18x^{2} + 4 - 6x^{2} + 7\).

Step 4: Combine Like Terms

Combine the terms with \(x^{2}\) and the constant terms separately. This will yield \( (18x^{2} - 6x^{2}) + (4 + 7) \) which simplifies to \(12x^{2} + 11\).

Step 5: Write the Final Simplified Expression

The expression in simplified form is \(12x^{2} + 11\).

Key Concepts

Distributive Property

The distributive property is a fundamental skill in algebra. It allows you to multiply a single term by each term within parentheses. This property can simplify expressions and make calculations manageable. For example, if you have an expression like \(6(x^2 - 2)\), the distributive property allows you to multiply 6 by each of the terms inside the brackets.

To apply the distributive property:

• Multiply 6 and \(x^2\) to get \(6x^2\).
• Then multiply 6 and -2 to get -12.
• Thus, \(6(x^2 - 2)\) simplifies to \(6x^2 - 12\).

Understanding the distributive property helps avoid mistakes in order of operations and is key in simplifying any complex algebraic expression.

Combining Like Terms

In algebra, simplifying an expression often involves combining like terms. Like terms have the same variable to the same power. For example, \(18x^2\) and \(6x^2\) are like terms because they both have \(x^2\).

To combine like terms effectively:

• Add or subtract the coefficients (numbers in front of the terms) of like terms. For \(18x^2 - 6x^2\), this means you subtract 6 from 18 to get \(12x^2\).
• Do the same for constant terms: when combining 4 and 7, you simply add them to get 11.

This helps to streamline the expression to its simplest form, which is crucial for further operations or setting equations.

Negative Sign Distribution

Handling negative signs correctly is crucial when simplifying expressions. When a negative sign precedes parentheses, it should be distributed to each term inside. This means changing the sign of each term within those brackets.

To distribute a negative sign effectively:

• Apply the negative sign as though it is a factor of -1. For instance, distributing a negative to \((6x^2 - 7)\) transforms it into \(-6x^2 + 7\).
• It's important to flip the sign of each term inside the parentheses.
  
  Thus, the original expression \(18x^2 + 4 - (6x^2 - 7)\) correctly becomes \(18x^2 + 4 - 6x^2 + 7\) after distributing the negative sign.

Remember, mastering the distribution of negative signs helps prevent errors and ensures that the expression is accurately simplified.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. This process often involves several steps such as distributing terms, handling negative signs, and combining like terms. The goal is to condense the expression into a form that is easier to work with for later calculations or solving conditions.

The final result, in our example, simplifies to \(12x^2 + 11\). Here's a quick recap of the steps:

• First, apply the distributive property.
• Next, simplify expressions within parentheses.
• Distribute any negative signs correctly.
• Lastly, combine like terms to arrive at the final expression.

Each of these steps is crucial to ensure that the algebraic expression is simplified correctly, facilitating further operations and better understanding of mathematical relationships.