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Zorluk: ZorQuadratic Equations

In the quadratic equation x2+bx+c=0x^2 + bx + c = 0, bb and cc are integers such that b+c=10b + c = 10. If the equation has no real solutions, what is the maximum possible value of cc?

  1. 18Cevap
  2. B
    19
  3. C
    14
  4. D
    6

Cevap

The maximum possible value of cc is 18.
For the quadratic equation to have no real solutions, the discriminant b2−4acb^2 - 4ac must be negative. Given a=1a=1, this simplifies to b2−4c<0b^2 - 4c < 0. Using the constraint b+c=10b + c = 10, we can express cc as 10−b10 - b. Substituting this into the inequality gives b2−4(10−b)<0b^2 - 4(10 - b) < 0, which simplifies to b2+4b−40<0b^2 + 4b - 40 < 0. Solving the boundary equation b2+4b−40=0b^2 + 4b - 40 = 0 using the quadratic formula gives b=−2±44b = -2 \pm \sqrt{44}. Since 44≈6.63\sqrt{44} \approx 6.63, the boundary values of bb are approximately −8.63-8.63 and 4.634.63. Thus, the inequality holds for −8.63<b<4.63-8.63 < b < 4.63. To maximize c=10−bc = 10 - b, we need the smallest integer value for bb in this range, which is b=−8b = -8. Substituting b=−8b = -8 gives the maximum value of c=10−(−8)=18c = 10 - (-8) = 18.

Adım Adım Çözüm

1
State the condition for the quadratic equation to have no real solutions.
The discriminant must be negative: b2−4ac<0b^2 - 4ac < 0. Since a=1a = 1, this simplifies to b2−4c<0b^2 - 4c < 0.
A quadratic equation has no real solutions if and only if its discriminant is strictly less than zero.
2
Express cc in terms of bb using the given equation b+c=10b + c = 10, and substitute it into the discriminant inequality.
Since c=10−bc = 10 - b, substituting gives b2−4(10−b)<0b^2 - 4(10 - b) < 0, which simplifies to b2+4b−40<0b^2 + 4b - 40 < 0.
Substituting one variable in terms of the other allows us to solve a single-variable quadratic inequality.
3
Find the boundary values of bb by solving the equation b2+4b−40=0b^2 + 4b - 40 = 0 using the quadratic formula.
b=−4±42−4(1)(−40)2=−4±1762=−2±44b = \frac{-4 \pm \sqrt{4^2 - 4(1)(-40)}}{2} = \frac{-4 \pm \sqrt{176}}{2} = -2 \pm \sqrt{44}. Since 44≈6.63\sqrt{44} \approx 6.63, the boundaries are approximately b≈−8.63b \approx -8.63 and b≈4.63b \approx 4.63.
The roots of the quadratic equation define the boundaries of the interval that satisfies the inequality.
4
Determine the interval for bb that satisfies the inequality and identify the value of bb that maximizes cc.
The inequality b2+4b−40<0b^2 + 4b - 40 < 0 is satisfied when −8.63<b<4.63-8.63 < b < 4.63. Since c=10−bc = 10 - b, maximizing cc requires selecting the smallest possible integer value of bb in this range, which is b=−8b = -8.
Subtracting a smaller number from 10 results in a larger value for cc.
5
Calculate the maximum value of cc using the optimal value of bb.
c=10−(−8)=18c = 10 - (-8) = 18.
Substituting the minimum integer value of bb gives the maximum integer value of cc.

Anahtar Kavram

Analyzing the discriminant of a quadratic equation to determine the nature of its roots and using substitution to solve optimization problems under constraints.
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Konular

Quadratic Equations

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