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Chapter 3: Problem 69

Find the value of \(x^{2}-3 x+1\) for each given value of \(x\). $$ -1 $$

Short Answer

Expert verified
The value is 5.

Step by step solution

01

Substitute the Given Value of x

We are given that \(x = -1\). Substitute \(-1\) into the expression \(x^2 - 3x + 1\). This gives us \((-1)^2 - 3(-1) + 1\).
02

Calculate the Square of x

Calculate \((-1)^2\). Since \(-1\) squared is \(1\), we have: \(1 - 3(-1) + 1\).
03

Multiply x by -3

Calculate \(-3(-1)\). Multiplying \(-3\) by \(-1\) gives \(+3\), so the expression becomes \(1 + 3 + 1\).
04

Add the Terms Together

Now, add \(1 + 3 + 1\) to find the final value. \(1 + 3 + 1 = 5\). Hence, the value of the expression \(x^2 - 3x + 1\) for \(x = -1\) is \(5\).

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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 technique used to evaluate algebraic expressions by replacing variables with specific values. This method is highly useful in both algebra and calculus for simplifying calculations.Key steps involve:
  • Identifying the variable and its given value in the expression.
  • Substitute the given value into the expression wherever the variable appears.
  • Perform arithmetic operations to simplify and find the result.

In our original exercise, we used the substitution method to find the value of the expression \(x^2 - 3x + 1\) when \(x = -1\). By substituting \(-1\) in place of \(x\), the expression becomes \((-1)^2 - 3(-1) + 1\). This transformed expression is then evaluated to determine the value.
Simplifying Expressions
Simplifying expressions involves performing arithmetic operations to rewrite an algebraic expression in its simplest form. This process makes it easier to work with the expression and reach solutions more efficiently.Important principles include:
  • Handling each mathematical operation step by step.
  • Combining like terms and reducing the expression to minimal terms.
  • Following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In our problem, after substitution, the expression \((-1)^2 - 3(-1) + 1\) needs to be simplified. We carried out the calculations as follows:- Firstly, \((-1)^2\) is calculated, resulting in \(1\).- Next, \(-3(-1)\) is worked out, giving \(+3\).- Finally, all terms \(1\), \(+3\), and \(1\) are added together resulting in \(5\). This is the simplest form of the expression for \(x = -1\).
Negative Numbers in Algebra
Working with negative numbers is a fundamental part of algebra. Negative numbers can initially pose challenges, but understanding how they interact in equations and expressions helps simplify complex problems.Key points to remember:
  • Squaring a negative number, such as \((-1)^2\), results in a positive number.
  • Multiplying two negative numbers, like \(-3\) and \(-1\), results in a positive number.
  • Be cautious of signs during operations; misplacing a negative sign can change the outcome entirely.

In the provided example, we demonstrated these principles:- When \((-1)^2\) was calculated, the result was \(1\).- The operation \(-3(-1)\) produced \(+3\) since multiplying two negatives gives a positive.This understanding allowed us to correctly evaluate the expression and reach the correct outcome. Managing negative numbers requires careful attention to ensure accuracy in algebraic calculations.

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Most popular questions from this chapter

Solve. See Example 4. The table shows the domestic box office (in billions of dollars) for the U.S. movie industry during the years shown. (Source: Motion Picture Association of America) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Box Office (in billions of dollars) } \\ \hline 2003 & 9.17 \\ \hline 2004 & 9.22 \\ \hline 2005 & 8.83 \\ \hline 2006 & 9.14 \\ \hline 2007 & 9.63 \\ \hline 2008 & 9.79 \\ \hline \end{array} $$ a. Write this paired data as a set of ordered pairs of the form (year, box office). b. In your own words, write the meaning of the ordered pair (2006,9.14) c. Create a scatter diagram of the paired data. Be sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

Answer each exercise with true or false. Point (3,0) lies on the \(y\) -axis.

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In 2000 , crude oil field production in the United States was 2130 thousand barrels. In \(2007,\) U.S. crude oil field production dropped to 1850 thousand barrels. (Source: Energy Information Administration) a. Write two ordered pairs of the form (years after 2000, crude oil production). b. Assume the relationship between years after 2000 and crude oil production is linear over this period. Use the ordered pairs from part (a) to write an equation of the line relating years after 2000 to crude oil production. c. Use the linear equation from part (b) to estimate crude oil production in the United States in 2010 , if this trend were to continue.

Graph each inequality. $$ 4 x+3 y \geq 12 $$

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2003,\) there were 302 million magazine subscriptions in the United States. By 2007 , this number was 322 million. (Source: Audit Bureau of Circulation, Magazine Publishers Association) a. Write two ordered pairs of the form (years after \(2003,\) millions of magazine subscriptions) for this situation. b. Assume the relationship between years after 2003 and millions of magazine subscriptions is linear over this period. Use the ordered pairs from part (a) to write an equation for the line relating year after 2003 to millions of magazine subscriptions. C. Use this linear equation in part (b) to estimate the millions of magazine subscriptions in 2005 .
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