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The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?
- a)9, 10
- b)10, 11
- c)11, 12
- d)12, 13
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The sum of the squares of two consecutive positive integers exceeds th...
Let the two consecutive positive integers be x and x + 1
x2 + (x + 1)2 - x(x + 1) = 91
x2 + x - 90 = 0
(x + 10)(x - 9) = 0 => x = -10 or 9.
As x is positive x = 9
Hence the two consecutive positive integers are 9 and 10.
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The sum of the squares of two consecutive positive integers exceeds th...
Problem:
The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?
Solution:
Let's assume the two consecutive positive integers as x and x+1.
Step 1: Set up the equation
According to the problem, the sum of the squares of two consecutive positive integers exceeds their product by 91. Therefore, we can set up the following equation:
x^2 + (x+1)^2 = x(x+1) + 91
Step 2: Simplify the equation
Expand the equation and simplify:
x^2 + x^2 + 2x + 1 = x^2 + x + 91
Combine like terms:
2x^2 + 2x + 1 = x^2 + x + 91
Step 3: Solve the equation
Move all terms to one side of the equation to solve for x:
2x^2 + 2x + 1 - x^2 - x - 91 = 0
Simplify further:
x^2 + x - 90 = 0
Factor the quadratic equation:
(x + 10)(x - 9) = 0
Set each factor equal to zero and solve for x:
x + 10 = 0 or x - 9 = 0
If x + 10 = 0, then x = -10, but we are looking for positive integers. Therefore, we discard this solution.
If x - 9 = 0, then x = 9.
Step 4: Find the consecutive integers
The two consecutive positive integers are x = 9 and x+1 = 10.
Therefore, the correct answer is option A) 9, 10.
The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?
Solution:
Let's assume the two consecutive positive integers as x and x+1.
Step 1: Set up the equation
According to the problem, the sum of the squares of two consecutive positive integers exceeds their product by 91. Therefore, we can set up the following equation:
x^2 + (x+1)^2 = x(x+1) + 91
Step 2: Simplify the equation
Expand the equation and simplify:
x^2 + x^2 + 2x + 1 = x^2 + x + 91
Combine like terms:
2x^2 + 2x + 1 = x^2 + x + 91
Step 3: Solve the equation
Move all terms to one side of the equation to solve for x:
2x^2 + 2x + 1 - x^2 - x - 91 = 0
Simplify further:
x^2 + x - 90 = 0
Factor the quadratic equation:
(x + 10)(x - 9) = 0
Set each factor equal to zero and solve for x:
x + 10 = 0 or x - 9 = 0
If x + 10 = 0, then x = -10, but we are looking for positive integers. Therefore, we discard this solution.
If x - 9 = 0, then x = 9.
Step 4: Find the consecutive integers
The two consecutive positive integers are x = 9 and x+1 = 10.
Therefore, the correct answer is option A) 9, 10.
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The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?a)9, 10b)10, 11c)11, 12d)12, 13e)None of theseCorrect answer is option 'A'. Can you explain this answer?
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