Question

One positive integer is 3 less than twice another. The sum of their squares is 698. Find the integers.

Answer 1

To find the two integers, we can set up two equations based on the given information. By substituting the value of x from the first equation into the second equation, we can solve for y. Then, substituting the value of y back into the first equation, we can find the value of x.

Step-by-step explanation:

To solve this problem, we can set up two equations based on the given information. Let's say the first integer is x and the second integer is y. We are given that x is 3 less than twice y, so we can write the equation: x = 2y - 3. We also know that the sum of their squares is 698, so the equation becomes x^2 + y^2 = 698. Now we can substitute the value of x from the first equation into the second equation and solve for y.

Substituting x = 2y - 3 into the equation x^2 + y^2 = 698, we get (2y - 3)^2 + y^2 = 698. Expanding this equation, we get 4y^2 - 12y + 9 + y^2 = 698. Combining like terms, we have 5y^2 - 12y + 9 = 698. Rearranging this equation and simplifying, we get 5y^2 - 12y - 689 = 0. Now we can solve this quadratic equation to find the value of y.

Using the quadratic formula, y = (-(-12) ± sqrt((-12)^2 - 4(5)(-689))) / (2(5)). Simplifying the equation further, we have y = (12 ± sqrt(144 + 13780)) / 10. Taking the positive value, y = (12 + sqrt(13924)) / 10. Evaluating this expression, we find y ≈ 9.7394. Now we can substitute this value back into the first equation to find x.

Using x = 2y - 3, we have x = 2(9.7394) - 3. Simplifying this equation, we get x ≈ 16.4788. Therefore, the two integers are approximately 16.4788 and 9.7394.

Answer 2

The value of positive integers are 21.22 and 12.11

Step-by-step explanation:

Given as :

The sum of squares of two integer = 698

Let The one positive integer be x

And The other positive integer be y

According to question

one positive integer = 3 less than twice the other positive integer

So, x = 2 × y - 3

I.e x = 2 y - 3

And x² + y² = 698

So, Put the value of x

I.e ( 2 y - 3 )² + y² = 698

or, 4 y² + 9 - 12 y + y² = 698

Or, 5 y² - 3 y - 698 = 0

Now solving this quadratic equation

y =
`\frac{-b\pm \sqrt{b^(2)-4* a* c}}{2* a}`

Or, y =
`\frac{3\pm \sqrt{-3^(2)-4* 5* -698}}{2* 5}`

Or, y =
`(3\pm √(13969))/(10)`

Or, y =
`(3\pm 118.19)/(10)`

∴ y = 12.11 , - 11.51

So , The value of y = 12.11

And the value of x = 2 × 12.11 - 3

I.e x = 21.22

Hence The value of positive integers are 21.22 and 12.11 Answer