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Chapter 11: Problem 12

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

Short Answer

Expert verified
The value of the expression is -1.

Step by step solution

01

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 \(x\):\[-(4)^{2} + 2(4) + 7\]
02

Calculate the square of x

Now, calculate the square of 4. \[(4)^{2} = 16\] So the expression becomes:\[-16 + 2(4) + 7\]
03

Multiply and simplify

Next, perform the multiplication: \[2(4)=8\]Hence, the expression is now:\[-16 + 8 + 7\]
04

Add and subtract in sequence

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Translate each phrase or sentence to a mathematical expression or equation. Ten times a quantity increased by two is nine.

Translate each phrase or sentence to a mathematical expression or equation. Twice a number is decreased by seven. This result is decreased by four times the number. The result is negative the original number, minus six.

For problems \(57-140\), solve each equation. $$ k=-4 $$

For problems \(57-140\), solve each equation. $$ \frac{2 x}{3}+1=5 $$

Translate each phrase or sentence to a mathematical expression or equation. A number is subtracted from nine. This result is then increased by one. The result is eight more than three times the number.
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