To solve the problem of when the baseball will hit the ground based on the quadratic function [tex]\( f(x) = -2x^2 + 3x + 5 \)[/tex], we can follow these steps:
1. Understand the Problem:
- The baseball’s height above the ground is modeled by the equation [tex]\( f(x) = -2x^2 + 3x + 5 \)[/tex], where [tex]\( x \)[/tex] represents the time in seconds.
- The baseball hits the ground when its height [tex]\( f(x) \)[/tex] is 0.
2. Set Up the Equation:
- We need to find the value of [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex].
- Set [tex]\( -2x^2 + 3x + 5 = 0 \)[/tex].
3. Apply the Quadratic Formula:
- The quadratic formula for solving [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- Here, [tex]\( a = -2 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 5 \)[/tex].
4. Calculate the Discriminant:
- The discriminant [tex]\(\Delta\)[/tex] (inside the square root) is calculated as:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
- Substituting the values:
[tex]\[
\Delta = 3^2 - 4(-2)(5) = 9 + 40 = 49
\][/tex]
5. Find the Roots:
- Using the quadratic formula, we get two possible values for [tex]\( x \)[/tex]:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-3 + \sqrt{49}}{2(-2)} = \frac{-3 + 7}{-4} = \frac{4}{-4} = -1
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-3 - \sqrt{49}}{2(-2)} = \frac{-3 - 7}{-4} = \frac{-10}{-4} = 2.5
\][/tex]
6. Interpret the Results:
- The two solutions for [tex]\( x \)[/tex] are [tex]\( -1 \)[/tex] and [tex]\( 2.5 \)[/tex]. However, since time [tex]\( x \)[/tex] cannot be negative, we discard [tex]\( -1 \)[/tex].
- The positive solution [tex]\( 2.5 \)[/tex] is the time in seconds when the baseball hits the ground.
Thus, the baseball will hit the ground after 2.5 seconds.
