Question

Add or subtract. $$\left(-5 x^{2}+2 x-12\right)-\left(6-9 x-7 x^{2}\right)$$

Short answer

The result of the calculation is \(2x^2 + 11x - 18\)

Step-by-step solution

Distribute Negative Sign

First, the negative sign in front of the second bracket should be distributed to all terms inside that bracket. Hence, \( - (6 - 9x - 7x^2) \) becomes \( -6 + 9x + 7x^2 \).

Combine Like Terms

Next, combine like terms from both the initial and adjusted brackets. The like terms combine as per the following operation \( -5x^2 + 7x^2 = 2x^2 \), \( 2x + 9x = 11x \), and \( -12 - 6 = -18 \)

Write the Final Result

Writing these results of the operation above, the final result yields \( 2x^2 + 11x - 18 \)

Key concepts

Combining Like Terms

In algebra, the process of combining like terms is fundamental for simplifying expressions. Like terms are terms in an algebraic expression that have the same variables raised to the same powers, even if their coefficients are different. For instance, in the given exercise, terms with the variable x squared (i.e. x²) or the variable x are considered like terms.

To combine like terms, we add or subtract the coefficients while keeping the variable part the same. For example,

• For the squared terms, -5x² and 7x², their combination results in 2x².
• For the linear terms in x, 2x and 9x, their sum is 11x.
• Lastly, the constants -12 and -6 combine to -18.

This is a crucial step in simplifying the polynomial to reach the final result of 2x² + 11x - 18.

Remember, it’s all about identifying and grouping the like terms, and then performing the arithmetic on the coefficients to simplify the expression.

Distributive Property

The distributive property is another essential concept in simplifying algebraic expressions. It allows you to remove parentheses by distributing a factor across terms within the parentheses. The property states that for any numbers or expressions a, b, and c, the equation a(b + c) is equal to ab + ac.

Applying this to our exercise, we see it in action when eliminating the negative sign before the second set of parentheses, transforming -(6 - 9x - 7x²) into -6 + 9x + 7x². Each term inside the parenthesis is multiplied by -1. This step is critical as it sets the stage for combining like terms correctly. Failure to apply the distributive property accurately can lead to errors in the final answer.

Example of Distributive Property

If we have 3(2x + 4), we distribute the 3 to both the 2x and the 4 separately, resulting in 6x + 12.

Algebraic Expressions

At the heart of this problem is the concept of algebraic expressions. An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Examples include 5x, -3y + 4, and 2x² - 11x + 5. Expressions can be simple, having just one term, or complex with many terms.

The exercise provided features a more complex expression, where the goal is to simplify by first applying the distributive property and then combining like terms. Understanding how to manipulate these expressions is key to solving problems in algebra.

Components of Algebraic Expressions

An algebraic expression is made up of:

• Constants: Numbers on their own, such as -12.
• Coefficients: Numbers that multiply the variables, like 5 in 5x.
• Variables: Symbols, usually letters, that represent unknown values.
• Exponents: Numbers that indicate how many times to multiply the variable by itself, as in x².
• Operation symbols: Including plus (+), minus (-), times (×), and divide (÷).

By learning to recognize these components and how to apply the distributive property and combine like terms appropriately, students can tackle a wide array of algebraic problems with confidence.