Question

solve for the zeros of the quadratic function $f(x) = x + 5 - 2x^2$.

1. put the quadratic function in standard form.

$f(x) = -2x^2 + x + 5$

1. set the function equal to 0 to create a quadratic equation.

$-2x^2 + x + 5 = 0$

1. determine the values for $a$, $b$, and $c$.

$a = $
$b = $
$c = $

1. analyze the discriminant.

$b^2 - 4ac = $
the quadratic function will have

Explanation:

Step1: Identify a, b, c

For quadratic equation \(ax^2 + bx + c = 0\), compare with \(-2x^2 + x + 5 = 0\). So \(a=-2\), \(b = 1\), \(c = 5\).

Step2: Calculate discriminant

Use formula \(b^2 - 4ac\). Substitute \(a=-2\), \(b = 1\), \(c = 5\):
\(1^2 - 4\times(-2)\times5 = 1 + 40 = 41\).
Since discriminant \(41>0\), the quadratic has two distinct real zeros.

Answer:

1. \(a = -2\), \(b = 1\), \(c = 5\)
2. \(b^2 - 4ac = 41\); The quadratic function will have two distinct real zeros.