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Question

Difficulty: MediumQuadratic Equations

In the quadratic equation x2+kx+36=0x^2 + kx + 36 = 0, kk is a constant. If the equation has two distinct real solutions, which of the following could be the value of kk?

  1. A
    −12-12
  2. −15-15Answer
  3. C
    −10-10
  4. D
    88

Answer

The value −15-15 is a possible value for kk because it is the only option that satisfies the inequality k<−12k < -12 or k>12k > 12, which guarantees a positive discriminant and thus two distinct real solutions.
A quadratic equation has two distinct real solutions when its discriminant is positive. For x2+kx+36=0x^2 + kx + 36 = 0, the discriminant is k2−4(1)(36)=k2−144k^2 - 4(1)(36) = k^2 - 144. The inequality k2−144>0k^2 - 144 > 0 simplifies to k2>144k^2 > 144, which means k>12k > 12 or k<−12k < -12. Among the choices, the value −15-15 is the only one that satisfies this condition.

Step-by-Step Solution

1
Identify the condition for a quadratic equation to have two distinct real solutions.
The discriminant, b2−4acb^2 - 4ac, must be strictly greater than 00.
A positive discriminant ensures the quadratic formula yields two distinct real solutions.
2
Substitute the coefficients of the given equation x2+kx+36=0x^2 + kx + 36 = 0 into the discriminant formula.
Here, a=1a = 1, b=kb = k, and c=36c = 36. The discriminant is k2−4(1)(36)=k2−144k^2 - 4(1)(36) = k^2 - 144.
This expresses the discriminant in terms of the constant kk.
3
Solve the inequality k2−144>0k^2 - 144 > 0 for kk.
k2>144⇒∣k∣>12k^2 > 144 \Rightarrow |k| > 12, which means k>12k > 12 or k<−12k < -12.
This gives the range of values for kk that produce two distinct real solutions.
4
Compare the given choices to the solution range.
Only −15-15 satisfies the condition because −15<−12-15 < -12.
This determines which of the options is a valid value for kk.

Key Concept

Determining the number of real solutions of a quadratic equation using the discriminant.

Alternative Method

Instead of solving the inequality algebraically, substitute each answer choice for kk into the equation and compute the discriminant b2−4acb^2 - 4ac directly to identify which option yields a positive value.
Estimated Time:1m 30s
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Topics

Quadratic Equations

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