SOLUTION: determine the value(s) of "k" for which there is one and only one real solution to the following quadratic: 10kx = -16x^2 - 25 Thank you

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Question 707046: determine the value(s) of "k" for which there is one and only one real solution to the following quadratic: 10kx = -16x^2 - 25
Thank you
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There will be one and only one real solution to a quadratic equation when the discriminant is zero. So let's start by putting the equation into standard form. Adding 16x%5E2 and 25 to each side:
16x%5E2%2B10kx%2B25=0
The discriminant, in general, is b%5E2-4ac. Our "a" is 16, "b" is 10k and "c" is 25. So our discriminant is:
%2810k%29%5E2-3%2816%29%2825%29
which simplifies to:
100k%5E2-1200
We want this to be zero so:
100k%5E2-1200+=+0
Now we solve for k. Dividing both sides by 100:
k%5E2-12+=+0
Adding 12:
k%5E2+=+12
So k+=+sqrt%2812%29 or k+=+-sqrt%2812%29. These square roots simplify:
k+=+sqrt%284%2A3%29 or k+=+-sqrt%284%2A3%29
k+=+sqrt%284%29%2Asqrt%283%29 or k+=+-sqrt%284%29%2Asqrt%283%29
k+=+2%2Asqrt%283%29 or k+=+-2%2Asqrt%283%29
These are the only possible values for k which will lead to one and only one real solution for x.