Problem 96 - College Algebra

Question

Simplify algebraic expression. \(14 x^{2}+5-\left[7\left(x^{2}-2\right)+4\right]\)

Step-by-Step Solution

Answer

\ The simplified form of the expression \(14 x^{2}+5-\left[7\left(x^{2}-2\right)+4\right]\) is \(7x^{2} + 15\)

Step 1: Distributive Property

Distribute the 7 across each term in the parenthesis: \(7 * x^{2}\) and \(7 * -2\) respectively. This yields a new expression: \(14 x^{2}+5-\left[7x^{2}-14+4\right]\)

Step 2: Combine Like Terms Inside Brackets

Combine -14 and 4 inside the square brackets to get -10. The resulting expression is : \(14x^{2}+5-\left[7x^{2}-10\right]\)

Step 3: Removing Brackets

Remove the bracket which requires changing the sign of each term within it. The expression thus becomes: \(14x^{2}+5-7x^{2}+10\)

Step 4: Combine Like Terms

Combine the like terms to simplify further: \(14x^{2} - 7x^{2}\) and \(5 + 10\). Therefore final simplified expression is \(7x^{2} + 15\)

Key Concepts

Distributive Property

The distributive property is all about multiplying a single term by each term inside a set of parentheses. This rule helps to eliminate parentheses and simplify expressions.
Let's take a closer look at a simple example to understand better.
Suppose you have the expression:
\( a(b + c) \).
You would use the distributive property by multiplying \( a \) with both \( b \) and \( c \).
So, \( a(b + c) \) becomes \( ab + ac \).
In the given exercise, the distributive property is applied to \( 7(x^2 - 2) \).

• Multiply 7 by \( x^2 \), resulting in \( 7x^2 \).
• Next, multiply 7 by \(-2\), resulting in \(-14\).

Thus, the expression is simplified using the distributive property to become \( 14x^2 + 5 - [7x^2 - 14 + 4] \).
By applying this property correctly, you can start simplifying more complex algebraic expressions.

Combining Like Terms

Combining like terms is a key concept in simplifying algebraic expressions.
Like terms are terms that have identical variable parts.
For example, in the expression \(3x + 2x\), both terms are "like" because they include the variable \(x\). Hence, they can be combined.
Back to our original expression. After using the distributive property, we had
\(14x^2 + 5 - [7x^2 - 14 + 4]\).
Inside the brackets, we have the numbers -14 and 4.
These are the constants and can be combined to make -10.
Thus, the expression becomes \(14x^2 + 5 - [7x^2 - 10]\).

• Look for all similar terms.
• Add or subtract them as needed.

Remember, this step is crucial for simplifying expressions further.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations.
They manage to express complex relationships in a simplified form.
In algebra, expressions can look overwhelming at first glance.
Breaking them down step-by-step helps make them more manageable.
Consider a typical expression,
like \( 4xy + 2x - 5 \).

• "4xy" is a term involving both x and y.
• "2x" involves only x.
• "-5" is a constant term.

Algebraic expressions never contain an equal sign, unlike equations.
You use them in various mathematical situations, from simple arithmetic to high-level calculus.
These expressions are simplified using techniques such as distributing, combining like terms, and removing parentheses.

Parentheses Removal

Removing parentheses is an essential skill in algebra.
It involves changing the signs of terms in the parentheses, especially after subtracting.
After combining like terms in the exercise, the expression was reduced to
\(14x^2 + 5 - [7x^2 - 10]\).
To remove the brackets, we need to ensure any subtraction is done correctly across the terms.

• Keep in mind that subtracting a negative is like adding a positive.
• This step simplifies expressions so that their terms can be effortlessly combined.

Applying this properly, the expression becomes \(14x^2 + 5 - 7x^2 + 10\).
Notice how the signs have changed when the brackets were opened.
This is crucial for maintaining the accuracy of your expression as you move forward in solving or simplifying it.