Problem 12 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-x^{2}+2 x+7, \text { if } x=4$$

Step-by-Step Solution

Answer

The value of the expression is -1.

Step 1: Substitute the value of x

The given expression is \[-x^{2} + 2x + 7\] We need to evaluate this expression for \(x = 4\). Substitute 4 in place of \(x\):\[-(4)^{2} + 2(4) + 7\]

Step 2: Calculate the square of x

Now, calculate the square of 4. \[(4)^{2} = 16\] So the expression becomes:\[-16 + 2(4) + 7\]

Step 3: Multiply and simplify

Next, perform the multiplication: \[2(4)=8\]Hence, the expression is now:\[-16 + 8 + 7\]

Step 4: Add and subtract in sequence

Start by adding and subtracting the terms in the expression:\[-16 + 8 = -8\] Now add 7:\[-8 + 7 = -1\]

Key Concepts

Substitution Method

The substitution method is a key mathematical approach used to evaluate algebraic expressions. Essentially, you replace a variable with a given number, allowing for numerical evaluation instead of working with abstract symbols. In our example, the expression

1. Start by identifying the variables in your expression. Here, the variable is \(x\).
2. Next, substitute \(x\) with the given value. For our exercise, we replace \(x\) with \(4\).
3. This substitution helps transform the original polynomial expression into a numerical one, making calculations straightforward.

Using the substitution method simplifies the expression evaluation process. With practice, you'll become proficient at solving equations that might have once seemed complex.

Polynomial Expressions

Polynomial expressions are fundamental concepts in algebra, consisting of variables, coefficients, and operations of addition, subtraction, and multiplication, but not division by a variable.

• In our exercise, we have the polynomial \(-x^2 + 2x + 7\).
• The expression includes different terms: \(-x^2\) is the quadratic term, \(2x\) the linear term, and \(7\) the constant term.

Each term in a polynomial can be separately evaluated to simplify the process:- **Quadratic term**: Requires squaring the variable.- **Linear term**: Involves multiplication of the variable by its coefficient.- **Constant term**: Remains unchanged.
Understanding these components aids in systematically evaluating and simplifying expressions.

Order of Operations

The order of operations is a critical rule set in mathematics dictating how to proceed with calculations involving more than one operation. The order is sometimes remembered using the acronym PEMDAS:

1. **P**arentheses
2. **E**xponents (like squaring a number)
3. **M**ultiplication and **D**ivision (from left to right)
4. **A**ddition and **S**ubtraction (from left to right)

In our algebraic expression, we see these concepts at work:- First, calculate the exponent, since \((-4)^2\) is resolved before any addition or subtraction.- Next, handle multiplications, like the \(2 \times 4\) in our example.- Finally, proceed with addition and subtraction sequentially across the expression.
This methodology ensures calculations are executed correctly and consistently for any algebraic expression.