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) = 4x^{2} + 2x + 2x + 1 = 4x^{2} + 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) = 4x^{2} + 6x + 6x + 9 = 4x^{2} + 12x + 9
Since the sum of the squares equal 202, we get
4x^{2} + 4x + 1 + 4x^{2} + 12x + 9 = 202
8x^{2} + 16x + 10 = 202
8x^{2} + 16x + 10 - 202 = 202 - 202
8x^{2} + 16x - 192 = 0
Divide everything by 8
We get x^{2} + 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 9^{2} + 11^{2} = 81 + 121 = 202
Aug 09, 16 01:40 PM
Average age of Dipu and Apu is 22 years. Average age of Dipu and Tipu is 24 years. Age of Dipu is 21 years. What are the ages of Apu, and tipu ? Let
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.