Final answer:
To find the zeros of the function k(x) = 6x^2 + 9x + 45, we attempt to factor the quadratic, but if it does not factor easily, we use the quadratic formula x = [-b ± √(b^2 - 4ac)]/(2a) to find the values of x that make the function equal zero.
Step-by-step explanation:
The student has asked to determine the zeros of the function k(x) = x2 + 5x2 + 9x + 45. First, let's clarify that there seems to be a typographical error in the expression. Presuming it is meant to read k(x) = 6x2 + 9x + 45, we can proceed to factor this quadratic equation.
Firstly we look for common factors, but in this case, there aren't any that will simplify the equation. The next step is to try and factor by grouping, but this equation does not easily lend itself to that method either. Given that the standard form of a quadratic equation is ax2 + bx + c = 0, this equation already appears to be in this form.
When the quadratic does not factor easily, we use the quadratic formula to determine the solutions for x. The quadratic formula is given by x = [-b ± √(b2 - 4ac)]/(2a), where a, b, and c are the coefficients from the quadratic equation ax2 + bx + c = 0.
The applicable values from the student's provided function would be a = 6, b = 9, and c = 45. Plugging these into the quadratic formula will provide the solutions for x, which are the zeros of the function. However, note that not all quadratics will have real number solutions; if the discriminant (b2 - 4ac) is negative, the solutions will be complex numbers.