Applications of quadratic equations
find two conservative odd integers such that the sum of their squares is 514

The two consecutive odd integers are -17 , -15 OR 15 , 17

Step-by-step explanation:

* Lets explain how to solve the problem

- The difference between each two consecutive odd integers is 2

- Assume that the first odd number is x, then the second odd integers

 is x + 2

∴ The two consecutive odd integers are x and x + 2

- The sum of their squares is 514

∵ The square of x is x²

∵ the square of x + 2 is (x + 2)²

∵ the sum of their squares is 514

∴ x² + (x + 2)² = 514

- Lets simplify it and solve it

∵ x² + (x + 2)² = 514

- Solve the bract of power 2

∵ (x + 2)² = (x)(x) + 2(x)(2) + (2)(2)

∴ (x + 2)² = x² + 4x + 4

∴ x² + x² + 4x + 4 = 514

- Add the like terms

∴ 2x² + 4x + 4 = 514

- Subtract 514 from both sides

∴ 2x² + 4x - 510 = 0

- All the terms have 2 common factor, then divide both sides by 2

∴ x² + 2x - 255 = 0

- Factorize the quadratic into two factors

∴ (x + 17)(x - 15) = 0

- Equate each factor by 0

∴ x + 17 = 0 OR x - 15 = 0

∵ x + 17 = 0

- Subtract 17 from both sides

∴ x = -17

- OR

∵ x - 15 = 0

- Add 15 to both sides

∴ x = 15

∵ x represents the first odd number

∴ x + 2 = -17 + 2 = -15  OR x + 2 = 15 + 2 = 17

∴ The two consecutive odd integers are -17 , -15 OR 15 , 17

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