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Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ?
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Let a, b be integers such that all the roots of the equation (x2 + ax ...
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Let a, b be integers such that all the roots of the equation (x2 + ax ...
Problem Analysis:
We are given an equation (x^2 - ax + 20)(a^2 - 17x + b) = 0 and we need to find the smallest possible value of a + b such that all the roots of the equation are negative integers.
Solution:
Step 1: Analyzing the First Quadratic Factor
Let's analyze the first quadratic factor: x^2 - ax + 20 = 0
The product of the roots of this quadratic equation is given by the constant term divided by the coefficient of the quadratic term. In this case, the product of the roots is 20/1 = 20.
Since we are looking for negative integer roots, the possible pairs of negative integer roots whose product is 20 are:
1) (-1, -20)
2) (-2, -10)
3) (-4, -5)
Step 2: Analyzing the Second Quadratic Factor
Now let's analyze the second quadratic factor: a^2 - 17x + b = 0
Similarly, the product of the roots of this quadratic equation is given by the constant term divided by the coefficient of the quadratic term. In this case, the product of the roots is b/1 = b.
Since we are looking for negative integer roots, the possible pairs of negative integer roots whose product is b are:
1) (-1, -b)
2) (-b, -1)
Step 3: Finding the Possible Values of a and b
To find the possible values of a and b, we need to combine the pairs of roots from the first and second quadratic factors.
Considering the first pair of roots for the first quadratic factor (-1, -20) and the first pair of roots for the second quadratic factor (-1, -b), we have:
a = (-1) + (-1) = -2
b = (-20) * (-1) = 20
Considering the second pair of roots for the first quadratic factor (-2, -10) and the second pair of roots for the second quadratic factor (-2, -b), we have:
a = (-2) + (-2) = -4
b = (-10) * (-2) = 20
Considering the third pair of roots for the first quadratic factor (-4, -5) and the first pair of roots for the second quadratic factor (-4, -b), we have:
a = (-4) + (-4) = -8
b = (-5) * (-4) = 20
Step 4: Finding the Smallest Value of a + b
We have found three possible values for a and b: (-2, 20), (-4, 20), (-8, 20).
The smallest possible value of a + b is -8 + 20 = 12.
Conclusion:
The smallest possible value of a + b is 12, which occurs when a = -8 and b = 20.
We are given an equation (x^2 - ax + 20)(a^2 - 17x + b) = 0 and we need to find the smallest possible value of a + b such that all the roots of the equation are negative integers.
Solution:
Step 1: Analyzing the First Quadratic Factor
Let's analyze the first quadratic factor: x^2 - ax + 20 = 0
The product of the roots of this quadratic equation is given by the constant term divided by the coefficient of the quadratic term. In this case, the product of the roots is 20/1 = 20.
Since we are looking for negative integer roots, the possible pairs of negative integer roots whose product is 20 are:
1) (-1, -20)
2) (-2, -10)
3) (-4, -5)
Step 2: Analyzing the Second Quadratic Factor
Now let's analyze the second quadratic factor: a^2 - 17x + b = 0
Similarly, the product of the roots of this quadratic equation is given by the constant term divided by the coefficient of the quadratic term. In this case, the product of the roots is b/1 = b.
Since we are looking for negative integer roots, the possible pairs of negative integer roots whose product is b are:
1) (-1, -b)
2) (-b, -1)
Step 3: Finding the Possible Values of a and b
To find the possible values of a and b, we need to combine the pairs of roots from the first and second quadratic factors.
Considering the first pair of roots for the first quadratic factor (-1, -20) and the first pair of roots for the second quadratic factor (-1, -b), we have:
a = (-1) + (-1) = -2
b = (-20) * (-1) = 20
Considering the second pair of roots for the first quadratic factor (-2, -10) and the second pair of roots for the second quadratic factor (-2, -b), we have:
a = (-2) + (-2) = -4
b = (-10) * (-2) = 20
Considering the third pair of roots for the first quadratic factor (-4, -5) and the first pair of roots for the second quadratic factor (-4, -b), we have:
a = (-4) + (-4) = -8
b = (-5) * (-4) = 20
Step 4: Finding the Smallest Value of a + b
We have found three possible values for a and b: (-2, 20), (-4, 20), (-8, 20).
The smallest possible value of a + b is -8 + 20 = 12.
Conclusion:
The smallest possible value of a + b is 12, which occurs when a = -8 and b = 20.
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Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ?
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Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ?.
Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a, b be integers such that all the roots of the equation (x2 + ax + 20)(a2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ?.
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