Question 1209497: 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 ikleyn(52778)   (Show Source):
You can put this solution on YOUR website! .
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]
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I just solved several similar problems yesterday and today, so at the regular studying process,
it is just a time for a visitor to turn on his own mind, having several TEMPLATES from me.
The given quadratic function 3-4k - (k+3)x - x^2 has the leading coefficient -1 at x^2,
so it represents a downward parabola.
In order for this quadratic polynomial be negative at all real values of x, the necessary and
sufficient condition is that the discriminant
d = b^2 - 4ac
be negative. Then the square function has no real roots and remains negative for all real values of x.
So, we write the discriminant
d = (-(k+3))^2 - 4*(-1)*(3-4k) = (k+3)^2 + 4*(3-4k) = k^2 + 6k + 9 + 12 - 16k = k^2 - 10k + 21.
It can be factored
d = (k-7)*(k-3).
So, the discriminant is negative
(k-7)*(k-3) < 0
if and only if the parameter "k" is between the roots 3 < k < 7,
when the factor (k-7) is negative, while the factor (k-3) is positive.
At this point, the problem is fully solved.
ANSWER. 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 is 3 < k < 7.
Solved.
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