Question

determine the number of solutions for the quadratic function f(x)=-x^2+3x-9

Answer 1

If solution in this situation means "how many different values of x will produce real values of y", then the answer is infinitely many as there is no restriction with polynomials on the values you may use for x.

If solution in this situation means "what are the roots of this quadratic function (found by factoring)", then there are always 2 for quadratic functions.

Answer 2

the choices are
no real solutions
1 solution (double root)
2 distinct real solutions

Answer 3

Okay that helps. So they are looking for you to factor the quadratic equation to find its roots. In this case, this function is not factorable as is, so we apply the all famous equation to find the roots:

\[x = \frac{ -b \pm \sqrt{b^{2}- 4ac} }{ 2a }\]If you substitute in, you should get the following values for x:
\[x = \frac{ 3 \pm \sqrt{3^{2}-4(-1)(-9)} }{ 2(-1) }\]\[x=\frac{ 3 \pm \sqrt{-27} }{ -2 }\]Therefore:
\[x1=\frac{ 3 + 3i \sqrt{3} }{ -2 }, x2=\frac{ 3 - 3i \sqrt{3} }{ -2 }\]
These roots, as you can see, involve i, which means they are complex. The answer you would give would be "there are no real solutions"

Answer 4

thank you so much!