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Difficulty: MediumQuadratic Equations and Factoring

The quadratic equation x2+bx+45=0x^2 + bx + 45 = 0, where bb is a constant, has two negative integer roots pp and qq such that p<qp < q. If q−p=4q - p = 4, what is the value of bb?

Answer: 14

Answer

The value of bb is 14.
For the quadratic equation x2+bx+45=0x^2 + bx + 45 = 0, the roots pp and qq must satisfy pq=45pq = 45 and p+q=−bp + q = -b. The negative integer factor pairs of 45 with p<qp < q are (−45,−1)(-45, -1), (−15,−3)(-15, -3), and (−9,−5)(-9, -5). Calculating the difference q−pq - p for each pair yields 44, 12, and 4, respectively. The condition q−p=4q - p = 4 uniquely identifies the roots as p=−9p = -9 and q=−5q = -5. Summing these roots gives p+q=−14p + q = -14, so b=−(−14)=14b = -(-14) = 14.

Step-by-Step Solution

1
Set up the relationships for the roots of the quadratic equation.
pq=45pq = 45 and p+q=−bp + q = -b.
For any quadratic equation x2+bx+c=0x^2 + bx + c = 0, the product of the roots equals cc and the sum of the roots equals −b-b.
2
Find all negative integer factor pairs (p,q)(p, q) of 45 such that p<qp < q.
The possible pairs are (−45,−1)(-45, -1), (−15,−3)(-15, -3), and (−9,−5)(-9, -5).
Since both roots are negative integers, their product is positive 45.
3
Determine the difference q−pq - p for each pair to match the given condition q−p=4q - p = 4.
For (−45,−1)(-45, -1), q−p=−1−(−45)=44q - p = -1 - (-45) = 44. For (−15,−3)(-15, -3), q−p=−3−(−15)=12q - p = -3 - (-15) = 12. For (−9,−5)(-9, -5), q−p=−5−(−9)=4q - p = -5 - (-9) = 4.
The pair (−9,−5)(-9, -5) satisfies q−p=4q - p = 4, establishing p=−9p = -9 and q=−5q = -5.
4
Calculate the coefficient bb.
b=−(p+q)=−(−9+(−5))=−(−14)=14b = -(p + q) = -(-9 + (-5)) = -(-14) = 14.
Substituting the root values into b=−(p+q)b = -(p + q) yields the final answer.

Key Concept

Factoring Quadratics and Relationships Between Roots and Coefficients
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Topics

Quadratic Equations and Factoring

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