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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.
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Root and StemDefinition of RootA root is an underground part of a plant that typically lacks nodes, leaves, and buds. It is responsible for anchoring the plant in the soil, absorbing water and minerals, and storing nutrients. Roots can be classified into two types: taproots and fibrous roots.Definition of StemA stem is the aboveground part of a plant that supports leaves, flowers, and fruits. It contains nodes, internodes, and buds. The primary functions of a stem are to transport water, nutrients, and sugars between the roots and the leaves, provide structural support, and facilitate the growth of the plant.Differences between Root and Stem1. Structure- Root: Roots are typically found underground and have no nodes or leaves. They have a central primary root, known as the taproot, from which smaller lateral roots branch out.- Stem: Stems are found aboveground and have nodes, internodes, leaves, and buds. They have a central main stem, known as the shoot, from which lateral branches and leaves arise.2. Function- Root: The main functions of roots are anchoring the plant in the soil, absorbing water and minerals, storing nutrients, and providing support.- Stem: Stems have several functions, including transporting water, minerals, and sugars between the roots and leaves, providing structural support, and facilitating growth and reproduction.3. Location- Root: Roots are typically found below the soil surface, although some plants have aboveground roots called aerial roots.- Stem: Stems are generally located above the ground, but they can also be partially or entirely underground.4. Growth- Root: Roots grow in length from the root tip and branches develop from the primary root.- Stem: Stems grow in height from the apical meristem located at the tip of the stem. This growth allows the plant to reach sunlight and maximize photosynthesis.5. Types- Root: Roots can be classified as taproots or fibrous roots based on their structure and branching pattern.- Stem: Stems can be classified as herbaceous or woody based on their texture and ability to undergo secondary growth.ConclusionIn summary, roots and stems are essential parts of a plant with distinct structures and functions. Roots are primarily responsible for anchoring the plant, absorbing water and minerals, and storing nutrients. Stems, on the other hand, support leaves, flowers, and fruits, transport substances throughout the plant, and provide structural support. Understanding the differences between roots and stems helps us appreciate the complexity and diversity of plant anatomy and physiology.

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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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