SOLUTION: How many solutions are there to the equation {{{f(x)=-0.2x^2+12x+11}}}? How do you know?

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Question 130302: How many solutions are there to the equation f%28x%29=-0.2x%5E2%2B12x%2B11? How do you know?

Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If you look at part c), you'll see that there are two solutions.


To figure out how many solutions a quadratic will have, simply use the discriminant

From the quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

the discriminant consists of all of the terms in the square root. So the discriminant is

D=b%5E2-4ac

the discriminant tells us how many solutions (and what type of solutions) we can expect for any quadratic.


Now let's find the discriminant for y=-0.2x%5E2%2B12x%2B11 (notice how a=-0.2, b=12, and c=11)


D=b%5E2-4ac Start with the given equation

D=%2812%29%5E2-4%2A%28-0.2%29%2A%2811%29 Plug in a=-0.2, b=12, c=11

D=144-4%2A%28-0.2%29%2A%2811%29 Square 12 to get 144

D=144%2B8.8 Multiply -4*(-0.2)*(11) to get 8.8

D=152.8 Add


Since the discriminant equals 152.8 (which is greater than zero) , this means there are two real solutions. Remember if the discriminant is greater than zero, then the quadratic will have two real solutions.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Your question is how many solutions, not how many real number solutions, or how many rational solutions. That makes the question very easy to answer. Just look at the highest exponent and that tells you how many solutions there are. For your problem, the highest exponent is 2, so there are exactly 2 roots. They aren't necessarily real number solutions, they could be complex numbers involving the imaginary number i which is defined by i%5E2=-1, but there are always the same number of solutions as the degree (highest exponent) of the equation.