Problem 66 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$3 x^{2}-\left[4 x^{2}-2 x-\left(x^{2}-2 x+6\right)\right]$$

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Identify the Inner Parentheses

The expression inside the inner parentheses is \( x^2 - 2x + 6 \). This expression needs to be simplified and removed first.

Step 2: Remove Inner Parentheses

Distribute the negative sign across the expression inside the inner parentheses: \(- (x^2 - 2x + 6) = -x^2 + 2x - 6\). Replace the inner parentheses in the original expression with this simplified expression.

Step 3: Simplify Expression with Removed Inner Parentheses

Substitute back into the expression: \[ 3x^2 - [4x^2 - 2x - x^2 + 2x - 6] \].Simplify the terms inside these brackets to get: \[ 4x^2 - x^2 = 3x^2 \] and the \(-2x + 2x = 0\) and the remaining term is \(-6\). So the expression is now \[ 3x^2 - [3x^2 - 6] \].

Step 4: Remove Brackets and Simplify Further

Distribute the negative sign: \( -[3x^2 - 6] = -3x^2 + 6 \).Substitute back into the expression: \( 3x^2 - 3x^2 + 6 \).

Step 5: Perform Final Simplification

The expression \( 3x^2 - 3x^2 + 6 \) simplifies to \( 0 + 6 \), which is simply \( 6 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, or division. They represent values and can be simplified, making them useful in solving equations and other mathematical tasks. In our exercise, the expression is given as \[ 3x^2 - \left[ 4x^2 - 2x - \left( x^2 - 2x + 6 \right) \right] \]. Understanding and simplifying expressions like these is essential in algebra.

• Variables: Symbols like \(x\) that represent unknown values.
• Terms: Parts of the expression that are added or subtracted, such as \(3x^2\) or \(-2x\).
• Coefficients: Numbers in front of variables, telling us how many of that variable are present, for example, \(3\) in \(3x^2\).

By mastering algebraic expressions, students can handle more complex equations with confidence and ensure their mathematical foundation is solid.

Parentheses Removal

Removing parentheses involves simplifying expressions by either performing operations inside the parentheses first or distributing any coefficients or negative signs. Parentheses can significantly change the outcome of calculations if not handled correctly. In our problem, we focus on removing both inner and outer parentheses.
First, we target the inner parentheses \( \left(x^2 - 2x + 6\right) \). We simplify this as \(- (x^2 - 2x + 6)\), resulting in \(-x^2 + 2x - 6\).
Then, we look at outer parentheses, expressed as \[ 3x^2 - \left[4x^2 - 2x - (-x^2 + 2x - 6)\right] \]. Simplifying what's inside helps us progressively remove these brackets.

• Inner parentheses removal: Always simplify expressions within before moving outward.
• Negative sign distribution: Be cautious when spreading a negative sign across terms within brackets.

With careful operation, removing parentheses becomes manageable and lays the groundwork for accurate computation.

Distributive Property

The distributive property is a crucial algebra skill allowing you to multiply a single term across terms inside parentheses. It simplifies expressions by removing brackets and combining like terms. In our example, it's indispensable for both distributing negatives and combining terms properly.

• For negative dispersal: Consider \(- (x^2 - 2x + 6) = -x^2 + 2x - 6\). The negative sign multiplies by each element inside.
• For bracket removal: When addressing \(-[3x^2 - 6] = -3x^2 + 6\), each element is again multiplied by \(-1\).

This property ensures each part of the expression is properly counteracted, particularly when brackets or parentheses are involved. It often works hand-in-hand with concepts like combining like terms and managing coefficients. Mastery of the distributive property is essential as it deals with breaking down complex equations and making them more solvable.