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Chapter 0: Problem 96

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

Short Answer

Expert verified
The simplified algebraic expression is \(7x^{2} + 15\).

Step by step solution

01

Simplify the Expression Inside the Square Brackets

Apply the distributive property \(a(b + c) = ab + ac\) to the term \(7(x^{2}-2)\) inside the square brackets. It simplifies to \(7x^{2} - 14\). Now reassemble the entire expression to get: \(14x^{2}+5-\left[7x^{2} -14 +4\right]\)
02

Simplify the Bracketed Term

Next, simplify inside the square brackets by combining like terms which yields:\(7x^{2} -10\). Place this result back into the original expression: \(14x^{2}+5-\left(7x^{2} -10\right)\)
03

Distribute the Negative Sign

Distribute the negative sign across each term within the parentheses. This changes the sign of each term and simplifies to: \(14x^{2}+5-(7x^{2} -10) = 14x^{2} +5 - 7x^{2} +10\}
04

Combine Like Terms

Finally, combine the like terms, which are terms that have the same variables and exponents. This results in the expression: \( (14x^{2} - 7x^{2}) + (5 +10) = 7x^{2} + 15\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra that is useful for simplifying expressions. This property states that when you multiply a single term by terms within a parenthesis, such as in the expression \(a(b + c) = ab + ac\), you must multiply the outside term by each term inside the parenthesis.
In the provided exercise, we needed to simplify the expression \(7(x^{2} - 2)\).
To do this, you apply the distributive property:
  • Multiply \(7\) by \(x^{2}\) to get \(7x^{2}\).
  • Then, multiply \(7\) by \(-2\) to yield \(-14\).
This gives us the expression \(7x^{2} - 14\).
Understanding how to use the distributive property makes it easier to break down and simplify complex expressions.
Combining Like Terms
Another key step in algebra simplification involves combining like terms. Like terms are terms in an expression that have the same variables raised to the same power.
In our example, after distributing and adjusting the terms, we're left to work within the expression \(7x^{2} - 14 + 4\).
We must combine like terms within the brackets:
  • Identify the numerical values without variables, such as \(-14\) and \(4\), and combine them to get \(-10\).
The simplified form is now \(7x^{2} - 10\).
In the broader expression \(14x^{2} + 5 - (7x^{2} - 10)\), we apply this combining of similar terms to simplify further.
Thus, by combining \(14x^{2}\) and \(7x^{2}\), we manage expression merging effectively, resulting in a cleaner formula.
Negative Sign Distribution
Distributing a negative sign across terms within parentheses is often perplexing but essential for correctly simplifying algebraic expressions.
Consider the bracketed expression \(-(7x^{2} - 10)\) in our problem.
Distributing the negative sign means changing the sign of each term inside:
  • Change \(7x^{2}\) to \(-7x^{2}\).
  • Change \(-10\) to \(+10\).
This leads us to the expression \(14x^{2} + 5 - 7x^{2} + 10\).
Once the negative sign has been properly distributed, you can easily proceed to combine like terms for a fully simplified result.
Thus, mastering the distribution of negative signs helps prevent common errors in simplification.

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Most popular questions from this chapter

What does it mean to solve a formula for a variable?

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

$$\begin{array}{l} \text { Find the exact value of } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}} \text { without } \\ \text { the use of a calculator. } \end{array}$$

Exercises \(142-144\) will help you prepare for the material covered in the next section. $$\text { Multiply: }\left(2 x^{3} y^{2}\right)\left(5 x^{4} y^{7}\right)$$

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3
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