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The difference of two positive integers is 3 and the sum of their squares is 89. Taking the smaller integer as x form a quadratic equation and solve it to find the integers. The integers are
- a)(7, 4)
- b)(5, 8)
- c)(3, 6)
- d)(2, 5)
Correct answer is option 'B'. Can you explain this answer?
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The difference of two positive integers is 3 and the sum of their squa...
Given information:
The difference of two positive integers is 3 (let's call them x and y).
The sum of their squares is 89.
Forming the equation:
Since the difference between x and y is 3, we can write an equation as follows:
x - y = 3
The sum of their squares is 89, so we can write another equation as follows:
x^2 + y^2 = 89
Solving the equation:
We can solve this system of equations by substitution or elimination method. Let's use the substitution method.
1. Rearrange the first equation to solve for x:
x = y + 3
2. Substitute this value of x into the second equation:
(y + 3)^2 + y^2 = 89
3. Expand and simplify the equation:
y^2 + 6y + 9 + y^2 = 89
2y^2 + 6y + 9 = 89
2y^2 + 6y - 80 = 0
4. Divide the equation by 2 to simplify it:
y^2 + 3y - 40 = 0
5. Factorize the quadratic equation:
(y + 8)(y - 5) = 0
6. Set each factor to zero and solve for y:
y + 8 = 0 or y - 5 = 0
7. Solve for y:
y = -8 or y = 5
8. Substitute the value of y back into the first equation to find x:
For y = -8:
x = -8 + 3 = -5 (not a possible solution since the question specifies positive integers)
For y = 5:
x = 5 + 3 = 8
Therefore, the two positive integers are 5 and 8, which matches option B in the given choices.
The difference of two positive integers is 3 (let's call them x and y).
The sum of their squares is 89.
Forming the equation:
Since the difference between x and y is 3, we can write an equation as follows:
x - y = 3
The sum of their squares is 89, so we can write another equation as follows:
x^2 + y^2 = 89
Solving the equation:
We can solve this system of equations by substitution or elimination method. Let's use the substitution method.
1. Rearrange the first equation to solve for x:
x = y + 3
2. Substitute this value of x into the second equation:
(y + 3)^2 + y^2 = 89
3. Expand and simplify the equation:
y^2 + 6y + 9 + y^2 = 89
2y^2 + 6y + 9 = 89
2y^2 + 6y - 80 = 0
4. Divide the equation by 2 to simplify it:
y^2 + 3y - 40 = 0
5. Factorize the quadratic equation:
(y + 8)(y - 5) = 0
6. Set each factor to zero and solve for y:
y + 8 = 0 or y - 5 = 0
7. Solve for y:
y = -8 or y = 5
8. Substitute the value of y back into the first equation to find x:
For y = -8:
x = -8 + 3 = -5 (not a possible solution since the question specifies positive integers)
For y = 5:
x = 5 + 3 = 8
Therefore, the two positive integers are 5 and 8, which matches option B in the given choices.
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The difference of two positive integers is 3 and the sum of their squares is 89. Taking the smaller integer as x form a quadratic equation and solve it to find the integers. The integers area)(7, 4)b)(5, 8)c)(3, 6)d)(2, 5)Correct answer is option 'B'. Can you explain this answer?
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