To determine where the acorn hits the ground, we need to solve the quadratic equation \( -2x^2 + 8x + 6 = 0 \).
First, we identify the coefficients in the quadratic equation:
[tex]\[
a = -2, \quad b = 8, \quad c = 6
\][/tex]
Next, we calculate the discriminant (\(\Delta\)) of the quadratic equation, which is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of \(a\), \(b\), and \(c\) into the formula:
[tex]\[
\Delta = 8^2 - 4(-2)(6) = 64 + 48 = 112
\][/tex]
Since the discriminant is positive (\(\Delta = 112\)), the quadratic equation has two real and distinct roots. The roots of the quadratic equation are found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values of \(a\), \(b\), and \(\Delta\):
[tex]\[
x = \frac{-8 \pm \sqrt{112}}{2(-2)} = \frac{-8 \pm \sqrt{112}}{-4}
\][/tex]
We simplify this further. Note that \(\sqrt{112}\) can be simplified as \(\sqrt{16 \cdot 7} = 4\sqrt{7}\):
[tex]\[
x = \frac{-8 \pm 4\sqrt{7}}{-4}
\][/tex]
Breaking this into two parts, we get the solutions:
[tex]\[
x_1 = \frac{-8 + 4\sqrt{7}}{-4} \quad \text{and} \quad x_2 = \frac{-8 - 4\sqrt{7}}{-4}
\][/tex]
Simplifying each:
[tex]\[
x_1 = \frac{-8 + 4\sqrt{7}}{-4} = 2 - \sqrt{7} \approx -0.65
\][/tex]
[tex]\[
x_2 = \frac{-8 - 4\sqrt{7}}{-4} = 2 + \sqrt{7} \approx 4.65
\][/tex]
Now we evaluate the reasonableness of the solutions in the context of the problem. Since \(x\) represents the horizontal distance from Jacob, it cannot be negative. Therefore:
- \(x = 2 - \sqrt{7} \approx -0.65\) feet is not a reasonable solution because distance cannot be negative.
- \(x = 2 + \sqrt{7} \approx 4.65\) feet is a reasonable solution because it represents a positive distance.
Thus, the acorn hits the ground approximately at \( x = 4.65 \) feet in front of Jacob, and the correct statement choice is:
A. [tex]\(x = 2 + \sqrt{7} \approx 4.65\)[/tex] feet and [tex]\(x = 2 - \sqrt{7} \approx -0.65\)[/tex] feet. Distance can't be negative, so [tex]\(x = 2 - \sqrt{7}\)[/tex] is not a reasonable solution.
