Problem 58 - Algebra 2

Question

REVIEW \(\left(-4 x^{2}+2 x+3\right)-3\left(2 x^{2}-5 x+1\right)=\) \(\mathbf{F} \quad 2 x^{2}\) \(\mathbf{G}-10 x^{2}\) \(\mathbf{H}-10 x^{2}+17 x\) \(\mathbf{J} \quad 2 x^{2}+17 x\)

Step-by-Step Solution

Answer

The answer is option H: -10x^{2} + 17x.

Step 1: Distribute the -3

The given expression is \((-4x^{2} + 2x + 3) - 3(2x^{2} - 5x + 1)\).First, distribute the \(-3\) inside the second set of parentheses:-3 times \(2x^{2}\) equals \(-6x^{2}\)-3 times \(-5x\) equals \(15x\)-3 times \(1\) equals \(-3\).This simplifies to \(-6x^{2} + 15x - 3\).

Step 2: Combine like terms

Now combine the terms from the expression \(-4x^{2} + 2x + 3 - 6x^{2} + 15x - 3\).Combine the \(x^2\) terms: \(-4x^{2} - 6x^{2} = -10x^{2}\).Combine the \(x\) terms: \(2x + 15x = 17x\).Combine the constant terms: \(3 - 3 = 0\).The simplified expression is \(-10x^{2} + 17x\).

Step 3: Match the simplified expression

The simplified expression is \(-10x^{2} + 17x\). Now, match this with the given multiple choice options:- \(\mathbf{F} \quad 2x^{2}\)- \(\mathbf{G} \quad -10x^{2}\)- \(\mathbf{H} \quad -10x^{2} + 17x\)- \(\mathbf{J} \quad 2x^{2} + 17x\)The correct answer is option \(\mathbf{H}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a number by a sum or difference inside parentheses. In simpler terms, it helps us "distribute" a multiplication operation over an addition or subtraction. This property is crucial in solving algebraic expressions and equations.
For example, in the expression \(a(b + c) \), the distributive property allows us to expand this to \(ab + ac\). This means that we multiply \(a\) with each term inside the parentheses individually.
In the context of the given problem, the expression \((-3)(2x^{2} - 5x + 1)\) utilizes the distributive property. We distribute \(-3\) to each term inside the parentheses:

• \(-3 \times 2x^{2} = -6x^{2}\)
• \(-3 \times (-5x) = 15x\)
• \(-3 \times 1 = -3\)

After distributing, these products combine to give \(-6x^{2} + 15x - 3\). This step is essential for simplifying the expression further while solving the exercise.

Combining Like Terms

Combining like terms is another important aspect in algebraic expressions. It involves combining terms that have the same variables raised to the same power. This simplifies expressions and makes them easier to work with.
In the problem we are solving, after distributing the \(-3\), we have the expression: \(-4x^{2} + 2x + 3 - 6x^{2} + 15x - 3\). To simplify this further, we need to combine like terms.
Here's how we do it:

• Combine the \(x^2\) terms: \-4x^{2}\ and \(-6x^{2}\) come together to form \(-10x^{2}\).
• Combine the \(x\) terms: \2x\ and \15x\ combine to yield \17x\.
• Combine the constant terms: \3\ and \(-3\) together become \0\.

After these combinations, the expression simplifies to \(-10x^{2} + 17x\), which is much more manageable to work with and identify in the multiple choice options.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They do not include an equal sign like equations do. Understanding how to work with algebraic expressions is essential in solving many math problems.
In the context of this problem, the expression \(-4x^{2} + 2x + 3 - 3(2x^{2} - 5x + 1)\) is an algebraic expression. It involves operations such as addition, subtraction, and multiplication, and includes terms with variables \(x^{2}\) and \(x\), as well as constant terms.
When working with such expressions:

• Identify key parts: terms, coefficients, variables, and constants.
• Apply properties like the distributive property to expand expressions.
• Combine like terms to simplify the expression.

These processes help transform complex expressions into a simpler form that is easier to analyze and solve. In this scenario, following these steps effectively led to finding the simplified expression \(-10x^{2} + 17x\), which matches with the correct option in the problem's choices.