Section 6.1 Order of Operations

What You Will Learn Order of Operations

Definitions Algebra: a generalized form of arithmetic Variables: letters used to represent numbers Constant: symbol that represents a specific quantity

Definitions Algebraic expression: a collection of variables, numbers, parentheses, and operation symbols

Definitions Equation: two algebraic expressions joined by an equal sign Solution to an Equation: the number or numbers that replace the variable to make the equation a true statement Solve an Equation: find the solution to an equation

Definitions Checking the solution: substitute the number for the variable in the equation If the result is a true statement, the number is a solution to the equation. If the result is a false statement, the number is not a solution. True: 1 is a solution

Definitions Evaluate an expression: find the value of the expression for a given variable 52, 2 is the exponent, 5 is the base “b to the nth power,” or bn, means

Order of Operations 1. First, perform all operations within parentheses or other grouping symbols (according to the following order). 2. Next, perform all exponential operations (that is, raising to powers or finding roots). 3. Next, perform all multiplications and divisions from left to right. 4. Finally, perform all additions and subtractions from left to right.

Example 1: Evaluating an Expression Evaluate the expression –x2 + 4x + 16 for x = 3. Solution –x2 + 4x + 16 = –(3)2 + 4(3) + 16 = –9 + 12 + 16 = 19

Example 2: Finding the Height A diver jumps from a platform diving board that is 32 feet above water. The height of the diver above water, in feet, t seconds after jumping from the platform, can be determined by the expression –16t2 + 20t + 32. Determine the diver’s height above the water 2 seconds after jumping from the platform.

Example 2: Finding the Height Solution –16t2 + 20t + 32 = –16(2)2 + 20(2) + 32 = –16(4) + 20(2) + 32 = –64 + 40 + 32 = –24 + 32 = 8 The diver is 8 feet above water 2 seconds after jumping from the platform.

Example 3: Substituting for Two Variables Evaluate –3x2 + 2xy – 2y2 when x = 2 and y = 3. Solution –3x2 + 2xy – 2y2 = –3(2)2 + 2(2)(3) – 2(9) = –3(4) + 2(2)(3) – 2(9) = –12 + 12 – 18 = –18

Example 4: Is 2 a Solution? Determine whether 2 is a solution to the equation 3x2 + 7x – 11 = 15. Solution 3x2 + 7x – 11 = 15 3(2)2 + 7(2) – 11 = 15 3(4) + 14– 11 = 15 12 + 14 – 11 = 15 26 – 11 = 15 15 = 15 True 2 is a solution to the equation.