find consecutive odd positive integers

by bibi
(city)

The sum of squares of two consecutive odd positive integers is 202. Find the integers.

Solution

Let 2x + 1 be the first odd integer

Then, 2x + 3 will be the second integer

Now square each odd integer

(2x + 1)2 = (2x + 1) × (2x + 1)


(2x + 1) × (2x + 1) = 2x × 2x + 2x × 1 + 1 × 2x + 1 × 1


(2x + 1) × (2x + 1) = 4x2 + 2x + 2x + 1 = 4x2 + 4x + 1


(2x + 3)2 = (2x + 3) × (2x + 3)


(2x + 3) × (2x + 3) = 2x × 2x + 2x × 3 + 3 × 2x + 3 × 3


(2x + 3) × (2x + 3) = 4x2 + 6x + 6x + 9 = 4x2 + 12x + 9


Since the sum of the squares equal 202, we get

4x2 + 4x + 1 + 4x2 + 12x + 9 = 202


8x2 + 16x + 10 = 202

8x2 + 16x + 10 - 202 = 202 - 202


8x2 + 16x - 192 = 0


Divide everything by 8

We get x2 + 2x - 24 = 0

To factor the left side of this equation, look for factors of -24 that will add up to 2.

6 × -4 = -24 and 6 + -4 = 2.

We get (x + 6) × (x + -4) = 0


We have two equation to solve

x + 6 = 0 and x + -4 = 0

x = -6 and x = 4

x = -6 cannot be used since the integers are positive

we can use x = 4.

The first integer is 2x + 1 = 2 × 4 + 1 = 9

The seond integer is 2x + 3 = 2 × 4 + 3 = 11

Indeed 92 + 112 = 81 + 121 = 202





Click here to post comments

Join in and write your own page! It's easy to do. How? Simply click here to return to Basic math word problem.




Recent Articles

  1. How to Derive the Equation of an Ellipse Centered at the Origin

    Mar 13, 19 11:50 AM

    Learn how to derive the equation of an ellipse when the center of the ellipse is at the origin.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

            Follow me on Pinterest


Math quizzes

 Recommended

Scientific Notation Quiz

Graphing Slope Quiz

Adding and Subtracting Matrices Quiz  

Factoring Trinomials Quiz 

Solving Absolute Value Equations Quiz  

Order of Operations Quiz

Types of angles quiz


Tough algebra word problems

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Recent Articles

  1. How to Derive the Equation of an Ellipse Centered at the Origin

    Mar 13, 19 11:50 AM

    Learn how to derive the equation of an ellipse when the center of the ellipse is at the origin.

    Read More

K-12 math tests


Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. 

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.