Question

Use a vertical format or a horizontal format to add or subtract. $$\left(-7 x^{2}+12\right)-\left(6-4 x^{2}\right)$$

Short answer

The simplified expression is \(-3 x^{2}+6\).

Step-by-step solution

Simplify the Expressions

The given expression is: \(-7 x^{2}+12-(6-4 x^{2})\). Normally, when subtracting the second expression from the first, we have to distribute the subtraction over each term in the expression being subtracted.

Distribute Subtraction

If we distribute the subtraction sign over the second expression, we get: \(-7 x^{2}+12-6+4 x^{2}\).

Combine Like Terms

Next, we combine like terms: Addition/subtraction of the independent terms and the terms with \(x^{2}\). This gives us: \((-7 x^{2}+4 x^{2})+(12-6)= -3 x^{2}+6\).

Key concepts

Combining Like Terms

Understanding how to combine like terms is essential in simplifying algebraic expressions. The term ‘like terms’ refers to terms that have the same variable raised to the same power. When you’re combining them, you simply add or subtract the coefficients - the numerical part of the terms - while keeping the variable part the same.

For example, in the expression ewline -7x^2 + 4x^2ewline, both terms have the variable 'x' raised to the second power. To combine these like terms, you subtract the coefficients because the second term is being subtracted. So, you compute ewline (-7) + (4) = -3ewline and keep the variable part unchanged. The result is ewline -3x^2ewline. In the case of independent terms (those without variables), like 12 and 6 in the exercise, you handle them the same way by performing the arithmetic operation of addition or subtraction

Why is this important?

Combining like terms is a foundational skill for manipulating and simplifying algebraic expressions. It allows you to condense expressions into their simplest form, making it easier to solve equations or understand the behavior of algebraic functions.

Distributive Property

The distributive property is a crucial algebraic property that allows you to multiply a single term across a sum or difference within parentheses. For subtraction, this means that when you have an expression like (-A) - (B - C), where A, B, and C are any algebraic expressions, you distribute the subtraction across the terms in the second expression to get -A - B + C.

In the provided exercise, the subtraction operation is distributed across the second expression, turning ewline 6 - 4x^2ewline into ewline -6 + 4x^2ewline. Observe that the sign of each term in the parentheses changes as a result of the distributive property. This is often a source of error for students who may forget to apply the sign change to each term.

Applying the Distributive Property

When you encounter a subtraction sign before parentheses, remember that you negate every term inside the parentheses. Think of it as multiplying the entire contents of the parentheses by -1. This step is a must before you can combine like terms in an expression.

Algebraic Expressions

Algebraic expressions represent mathematical relationships using variables, constants, and arithmetical operations. These expressions are the building blocks for forming equations and solving various algebraic problems.

The expression from the exercise, ewline -7x^2 + 12 - (6 - 4x^2)ewline, is a combination of terms that includes constants like 12 and 6, and variable terms like ewline -7x^2ewline and ewline 4x^2ewline. The goal when working with algebraic expressions is to manipulate them in ways that simplify or solve them, such as combining like terms or applying properties such as the distributive property.

Variables and Constants

In algebraic expressions, variables are symbols that represent unknown values and constants are fixed values. Variables are important when forming general solutions, and constants provide specific information that often relates to real-world quantities. Understanding both is crucial for the effective manipulation of algebraic expressions in diverse mathematical contexts.