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Chapter 9: Problem 68

Evaluate each expression for \(x=3\). $$ (x-1)^{2} $$

Short Answer

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Step by step solution

01

Substitute the given value of x

Replace the variable x with 3 in the expression \( (x-1)^2 \). This gives us \( (3-1)^2 \).
02

Simplify inside the parentheses

Simplify the expression inside the parentheses first: \( 3-1 = 2 \). The expression now becomes \( 2^2 \).
03

Evaluate the power

Raise 2 to the power of 2: \( 2^2 = 4 \).

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Key Concepts

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

Substitute Variable
In algebra, one of the first steps to solving many problems is to substitute the variable with a given number. This simply means replacing the variable, often represented by a letter like \( x \), with the number provided in the problem. Let鈥檚 take a closer look using our exercise as an example. Given \( x=3 \), we replace \( x \) in the expression \( (x-1)^2 \) with 3. This transforms our expression into \( (3-1)^2 \). Doing this helps convert the algebraic expression into a numerical one, making it easier to work with.
Simplify Expressions
Once you have substituted the variable, the next crucial step is to simplify the expression. Simplification involves performing all the arithmetic calculations needed to reduce the expression. This often involves the terms within parentheses and other basic operations like addition, subtraction, multiplication, and division. For our exercise, after substitution, we have the expression \( (3-1)^2 \). We simplify inside the parentheses first, where \( 3-1 \) becomes 2, turning our expression into \( 2^2 \). Following the right order of operations, commonly remembered by PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction), ensures accurate results.
Exponentiation
The final concept we need to understand here is exponentiation, which involves raising a number to the power of another number. In our exercise, the simplified expression is \( 2^2 \). Here, 2 is the base, and the exponent is also 2. Exponentiation tells us to take the base number and multiply it by itself as many times as the exponent indicates. For \( 2^2 \), this means multiplying 2 by itself: \( 2 \times 2 = 4 \). It's a fundamental concept in algebra that often helps simplify expressions further and achieve the final numerical result.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ 2 x^{2}=5-2 x $$

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ x^{2}=1+x $$

Use a calculator with a square root key to solve each equation. Round your answers to the nearest hundredth. \((k+2.14)^{2}=5.46\)

Write each quotient in standard form. $$ \frac{-i}{1+2 i} $$

Solve each problem. An astronaut on the moon throws a baseball upward. The altitude (height) \(h\) of the ball, in feet, \(x\) seconds after he throws it, is given by the equation $$ h=-2.7 x^{2}+30 x+6.5 $$ At what times is the ball 12 ft above the moon's surface?
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