Question 1209534: Find the range of values of k for which the expression 3 - 4K - (k+3)x - x^2 will be negative for all real values of x.
[the answer is 3 < k < 7]
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the range of values for k:
**1. Rewrite the expression:**
The given expression is 3 - 4k - (k+3)x - x². We can rewrite this as:
-x² - (k+3)x + (3 - 4k)
**2. Consider the quadratic form:**
This is a quadratic expression in x of the form ax² + bx + c, where:
* a = -1
* b = -(k+3)
* c = (3 - 4k)
**3. Condition for negativity:**
For the quadratic expression to be negative for all real values of x, two conditions must be met:
* **The parabola must open downwards:** This is already satisfied since a = -1 (negative).
* **The quadratic must have no real roots:** This means the discriminant (b² - 4ac) must be less than zero.
**4. Apply the discriminant condition:**
The discriminant is:
b² - 4ac = [-(k+3)]² - 4*(-1)*(3 - 4k)
= (k² + 6k + 9) + 4(3 - 4k)
= k² + 6k + 9 + 12 - 16k
= k² - 10k + 21
For no real roots, the discriminant must be less than zero:
k² - 10k + 21 < 0
**5. Solve the inequality:**
Factor the quadratic:
(k - 3)(k - 7) < 0
This inequality is satisfied when k is between the roots 3 and 7.
**6. Final answer:**
Therefore, the expression 3 - 4k - (k+3)x - x² will be negative for all real values of x when:
3 < k < 7
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