Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 12
An integer is 5 more than another. If the product of the two integers is equal to 2 more than twice their sum, then find the integers.
Short Answer
Expert verified
The integers are (3, 8) and (-4, 1).
Step by step solution
01
Define Variables
Let the two integers be represented as \( x \) and \( y \). Let \( x \) be the integer that is 5 more than \( y \), so we can write the equation \( x = y + 5 \).
02
Write Equation for Product
The product of the two integers is given as \( xy \). Substituting for \( x \), we get \( (y + 5)y = y^2 + 5y \).
03
Write Equation for Twice the Sum Plus Two
The sum of the two integers is \( x + y = (y + 5) + y = 2y + 5 \). Twice the sum plus two is \( 2(2y + 5) + 2 = 4y + 10 + 2 = 4y + 12 \).
04
Set Product Equal to Twice the Sum Plus Two
We know that \( xy = 4y + 12 \). Substitute \( xy \) from Step 2: \( y^2 + 5y = 4y + 12 \).
05
Simplify and Solve for \( y \)
Rearrange the equation to set it to zero: \( y^2 + 5y - 4y - 12 = 0 \), which simplifies to \( y^2 + y - 12 = 0 \). This is a quadratic equation.
06
Factor the Quadratic Equation
Factor \( y^2 + y - 12 \). It factors into \( (y - 3)(y + 4) = 0 \).
07
Find Possible Values for \( y \)
Set each factor to zero: \( y - 3 = 0 \) gives \( y = 3 \), and \( y + 4 = 0 \) gives \( y = -4 \).
08
Calculate Corresponding \( x \) Values
For \( y = 3 \), \( x = y + 5 = 3 + 5 = 8 \). For \( y = -4 \), \( x = y + 5 = -4 + 5 = 1 \). Thus the possible integer pairs are \((3, 8)\) and \((-4, 1)\).
09
Verify Solutions
For \( (3, 8) \), calculate \( 3 \times 8 = 24 \) and \( 2(3 + 8) + 2 = 24 \). Matches. For \( (-4, 1) \), calculate \( -4 \times 1 = -4 \) and \( 2(-4 + 1) + 2 = -4 \). Matches.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
Quadratic equations are a type of equation where the highest power of the variable is squared, typically in the form of \( ax^2 + bx + c = 0 \). These equations are fundamental in algebra and appear frequently in various math problems.
Recognizing and solving these equations is essential for mastering algebraic techniques and understanding more complex problems.
- The standard form \( ax^2 + bx + c = 0 \) represents a parabola when graphed on a coordinate plane.
- To solve a quadratic equation, you can factor it, complete the square, or use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- In this exercise, after simplifying our algebraic expression, we derived a quadratic equation: \( y^2 + y - 12 = 0 \).
Recognizing and solving these equations is essential for mastering algebraic techniques and understanding more complex problems.
Factoring Polynomials
Factoring polynomials is a method to express a polynomial equation as a product of its simpler polynomials. Particularly useful for solving quadratic equations, factoring involves breaking down the polynomial into expressions that multiply to form the original polynomial.
- In the case of a quadratic equation like \( y^2 + y - 12 = 0 \), factoring involves finding two numbers that multiply to \(-12\) (the constant term) and add to \(1\) (the coefficient of the linear term \(y\)).
- These two numbers are \(3\) and \(-4\), so the factored form is \((y - 3)(y + 4) = 0 \).
- Factoring allows us to apply the zero-product property, which states that if a product of two factors is zero, at least one of the factors must be zero. Therefore, \(y - 3 = 0\) and \(y + 4 = 0\).
Algebraic Equations
Algebraic equations are statements of equality between two expressions involving variables. These equations are the backbone of algebra as they help describe relationships between quantities.
- When working with algebraic equations, the goal is often to find the unknown variable or variables that satisfy the equation.
- In this problem, we formulated an equation based on given conditions: two integers add up, their product, and some transformations to this sum. This led to an equation that required solving for \( y \).
- By isolating the variable, first by substitution and then simplifying, we converted the equation into a quadratic form, allowing us to solve it through factoring.
