Problem 43
Question
\(\left(3 x^{2}-15\right)+(4 x-9)\)
Step-by-Step Solution
Verified Answer
The simplified expression is \(3x^2 + 4x - 24\).
1Step 1: Simplify Each Group
First, we'll simplify each group separately. The first group is \(3x^2 - 15\) and the second one is \(4x - 9\).
2Step 2: Combine Like Terms
Next, combine the expressions \(3x^2 - 15\) and \(4x - 9\). Since there are no like terms between these two groups, the final expression remains as \(3x^2 - 15 + 4x - 9\).
3Step 3: Simplify Constant Terms
Simplify the constant terms from the combined expression: \(-15 - 9 = -24\). So the simplified expression is \(3x^2 + 4x - 24\).
Key Concepts
Combining Like TermsSimplifying ConstantsGrouping Terms
Combining Like Terms
When simplifying algebraic expressions, it is crucial to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(6x + 3x + 4\), both \(6x\) and \(3x\) are like terms because they both have the variable \(x\) to the first power. To combine them, simply add their coefficients: \(6 + 3 = 9\), resulting in \(9x\).
It's important to keep an eye on the coefficients and powers of the variables. Always ensure they match before combining them. Knowing how to identify and combine like terms helps to simplify complicated expressions, making them more manageable.
It's important to keep an eye on the coefficients and powers of the variables. Always ensure they match before combining them. Knowing how to identify and combine like terms helps to simplify complicated expressions, making them more manageable.
Simplifying Constants
Simplifying constants involves dealing with numbers that do not have variables attached to them. In the given exercise, we see the constants \(-15\) and \(-9\). To simplify, add or subtract these constants like regular numbers. Here, you would combine \(-15\) and \(-9\):
\[ -15 - 9 = -24 \]
This step reduces the expression to a simpler form. Simplifying constants is key because it minimizes the number of terms, making further calculations easier.
\[ -15 - 9 = -24 \]
This step reduces the expression to a simpler form. Simplifying constants is key because it minimizes the number of terms, making further calculations easier.
Grouping Terms
Grouping terms involves reorganizing an expression to make it easier to work with and simplify. You often group similar terms together: like terms with each other and constants with each other. Let's consider the given example. We'll start with the original expression: \((3x^2 - 15) + (4x - 9)\).
First, treat each group separately. For the groups \((3x^2 - 15)\) and \((4x - 9)\), simplify if possible. In this example, there are no immediate simplifications within each group, so we move on to combining them.
When combining, observe the terms. Since there are no like terms, just write them together: \((3x^2 + 4x - 15 - 9)\). Next, combine the constants: \(-15 - 9 = -24\). The final simplified expression is: \((3x^2 + 4x - 24)\).
First, treat each group separately. For the groups \((3x^2 - 15)\) and \((4x - 9)\), simplify if possible. In this example, there are no immediate simplifications within each group, so we move on to combining them.
When combining, observe the terms. Since there are no like terms, just write them together: \((3x^2 + 4x - 15 - 9)\). Next, combine the constants: \(-15 - 9 = -24\). The final simplified expression is: \((3x^2 + 4x - 24)\).