Textbook Question
Use Choices A–D to answer each question. A. 3x^2 - 17x - 6 = 0 B. (2x + 5)^2 = 7 C. x^2 + x = 12 D. (3x - 1)(x - 7) = 0 Which equation is set up for direct use of the square root property? Solve it
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1. Equations & Inequalities
The Square Root Property
3:42 minutesProblem 18
Textbook Question
Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. The difference of the squares of two positive consecutive odd integers is 32. Find the integers.
Verified step by step guidance1
Let the first odd integer be represented by \( x \).
The next consecutive odd integer can be represented by \( x + 2 \).
The square of the first integer is \( x^2 \) and the square of the second integer is \((x + 2)^2\).
Set up the equation for the difference of the squares: \((x + 2)^2 - x^2 = 32\).
Simplify and solve the equation: Expand \((x + 2)^2\) to get \(x^2 + 4x + 4\), then subtract \(x^2\) and solve for \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Consecutive Integers
Consecutive integers are numbers that follow each other in order without any gaps. For example, if x is an integer, then x and x+1 are consecutive integers. This concept is crucial for understanding how to represent and manipulate sequences of numbers, particularly when dealing with odd and even integers.
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Difference of Squares
The difference of squares is a mathematical identity that states a² - b² = (a - b)(a + b). This concept is essential for solving problems involving the squares of numbers, as it allows for the simplification of expressions and the solving of equations. In this context, it helps to relate the two consecutive odd integers to their difference.
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Odd and Even Integers
Odd integers are numbers that cannot be divided evenly by 2, while even integers can. For example, if x is an odd integer, then x+2 will also be odd. Understanding the properties of odd and even integers is vital for solving problems that involve their relationships, especially when determining consecutive odd integers and their squares.
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