Problem 74 - Algebra 1: Concepts and Skills

Question

Evaluate the expression when \(x=-2\) (Lessons \(1.3,2.3,2.5)\). $$ x^{2}+7 x+9 $$

Step-by-Step Solution

Answer

The result when \(x=-2\) in the given polynomial is \(-1\).

Step 1: Substitute the value of x

We'll start by substituting \(x = -2\) into the expression \(x^{2}+7 x+9\), so it becomes \((-2)^{2}+7(-2)+9\)

Step 2: Execute exponentiation

Next, crunch \((-2)^{2}\) to get 4. After simplification, the expression will be \(4+7(-2)+9\).

Step 3: Perform multiplication

Then perform the multiplication in \(7(-2)\) to get -14. Therefore, our equation simplifies to \(4-14+9\).

Step 4: Perform addition and subtraction from left to right

Finally, perform the subtraction and addition from left to right to get the result: \(-1\).

Key Concepts

Substitution in Algebra

Substitution is a fundamental technique in algebra that involves replacing a variable with a given number. Here, we substituted the value \(x = -2\) into the polynomial expression \(x^2 + 7x + 9\).

• Start by replacing every instance of the variable \(x\) in the expression with \(-2\). This gives us \((-2)^2 + 7(-2) + 9\).
• Substitution sets the stage for simplifying the expression. It's like plugging in a specific number to find out what the expression evaluates to.
• This process transforms abstract expressions into concrete numbers, making further calculations possible and straightforward.

By substituting the given value of \(x\), we change the expression solely into a numeric calculation that can be further simplified. This allows us to determine the value of the expression for that specific number.

Order of Operations

When simplifying mathematical expressions, the order in which operations are performed is crucial. This is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

For the expression \((-2)^2 + 7(-2) + 9\):
The first operation is to evaluate the exponent \((-2)^2\), which simplifies to \(4\). Thus, the expression becomes \(4 + 7(-2) + 9\).
Next, execute the multiplication of \(7\) and \(-2\), resulting in \(-14\). Hence, the expression now reads \(4 - 14 + 9\).
Finally, perform addition and subtraction from left to right by first calculating \(4 - 14\) to get \(-10\), and then adding \(9\) to find the final result, \(-1\).
This ordered approach ensures that everyone arrives at the correct answer consistently, by following a universal set of rules for simplifying expressions.

Polynomial Evaluation

Evaluating a polynomial means finding the value of a polynomial expression for a specific value of the variable. Here, we evaluated the polynomial \(x^2 + 7x + 9\) at \(x = -2\).

• A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, such as \(x^2 + 7x + 9\).
• To evaluate, substitute the given value of the variable. In our case, replace \(x\) with \(-2\).
• After substitution, follow the order of operations to simplify the expression to a single numerical value.

Evaluating polynomials is a common task that allows us to understand the behavior and value of polynomial functions at specific points. This helps in various applications in science, engineering, and finance, where functions need to be evaluated for different inputs to predict or explain phenomena.