Final answer:
To find the zeros and factors of the quadratic function g(x) = 2x^2 + 9x - 5, one can use the quadratic formula, yielding zeros at x = 1/2 and x = -5. The quadratic does not factor neatly over the integers, but an approximate factored form, based on the zeros found, would be (2x - 1)(x + 5).
Step-by-step explanation:
The factors and zeros of the quadratic function g(x) = 2x^2 + 9x - 5 can be found using the quadratic formula, which is applicable to any quadratic equation of the form ax^2 + bx + c = 0. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a). In this case, a = 2, b = 9, and c = -5.
To find the zeros of the function, we apply the quadratic formula as follows:
- x = (-(9) ± √((9)^2 - 4(2)(-5))) / (2(2))
- x = (-9 ± √(81 + 40)) / 4
- x = (-9 ± √(121)) / 4
- x = (-9 ± 11) / 4
Therefore, the zeros of g(x) are x = 1/2 and x = -5, which are the solutions to the equation g(x) = 0.
Factoring the quadratic expression to find its factors is equivalent to rewriting g(x) as a product of two binomials. However, this quadratic expression does not factor easily over the integers, so the factored form would involve the solutions we found using the quadratic formula. Hence, an approximate factored form is (2x - 1)(x + 5).