Problem 42 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 4 x 2+13 x+36+3 x 2+6 x-27 $$

Step-by-Step Solution

Answer

The simplified expression is \(7x^2 + 19x + 9\).

Step 1: Identify Like Terms

Recognize the like terms in the expression. The expression given is \(4x^2 + 13x + 36 + 3x^2 + 6x - 27\). The like terms are those with the same variable and exponent.- Quadratic terms: \(4x^2\) and \(3x^2\).- Linear terms: \(13x\) and \(6x\).- Constant terms: \(36\) and \(-27\).

Step 2: Combine Quadratic Terms

Add together the quadratic terms. \[4x^2 + 3x^2 = 7x^2\]

Step 3: Combine Linear Terms

Add together the linear terms.\[13x + 6x = 19x\]

Step 4: Combine Constant Terms

Add together the constant terms.\[36 + (-27) = 9\]

Step 5: Write the Simplified Expression

Combine all the results from the previous steps to form the simplified expression: \[7x^2 + 19x + 9\]

Key Concepts

Like Terms

In algebra, the phrase "like terms" refers to terms in an expression that have the same variable raised to the same power. Identifying and combining like terms is an essential step in simplifying algebraic expressions.
When simplifying, focus on grouping terms that look alike:

• Terms with the same variables and exponents are like terms.
• You can only combine these terms through addition or subtraction.

For example, in the expression \(4x^2 + 3x^2 + 13x + 6x + 36 - 27\), the like terms are \(4x^2\) and \(3x^2\), \(13x\) and \(6x\), and the constants \(36\) and \(-27\). Recognizing and combining them will simplify the expression.

Quadratic Terms

Quadratic terms are terms in an algebraic expression where the variable is raised to the power of two. These are usually represented as \(ax^2\), where \(a\) is a coefficient. Quadratic terms are crucial in expressions, especially when solving quadratic equations.
In the exercise \(4x^2 + 3x^2 + 13x + 6x + 36 - 27\), the quadratic terms are \(4x^2\) and \(3x^2\).
To simplify, add these together:

• \(4x^2 + 3x^2 = 7x^2\)

This new term, \(7x^2\), is the simplified form of the quadratic terms combined.

Linear Terms

Linear terms are components of expressions with a variable raised to the power of one. They look like \(bx\), where \(b\) is the coefficient. These terms follow straight line equations when graphed.
For the given expression \(4x^2 + 3x^2 + 13x + 6x + 36 - 27\), the linear terms are \(13x\) and \(6x\).
To simplify these, simply add them together:

• \(13x + 6x = 19x\)

By combining these linear terms, you get \(19x\), which is an integral part of the simplified result.

Constant Terms

Constant terms are parts of an expression that do not contain variables. These remain the same regardless of the value of the variable and are represented as fixed numbers. In any expression, you can think of them as the standalone numbers.
In \(4x^2 + 3x^2 + 13x + 6x + 36 - 27\), the constant terms are \(36\) and \(-27\).
To simplify, combine these constants through addition or subtraction:

• \(36 + (-27) = 9\)

After adding the constant terms, \(9\) becomes part of the simplified form of the expression.