Questions: Find the zeros of the function (f(x)=2 x^2+13.5 x+17.4). Round values to the nearest thousandths (if necessary).

Solution Steps

To find the zeros of the function \( f(x) = 2x^{2} + 13.5x + 17.4 \), we first calculate the discriminant \( D \) using the formula:

\[ D = b^2 - 4ac \]

Substituting the values \( a = 2 \), \( b = 13.5 \), and \( c = 17.4 \):

\[ D = (13.5)^2 - 4 \cdot 2 \cdot 17.4 = 182.25 - 139.2 = 43.05 \]

Since the discriminant \( D \) is positive (\( D > 0 \)), we can find two distinct real roots using the quadratic formula:

\[ x = \frac{{-b \pm \sqrt{D}}}{2a} \]

Using the positive square root for the first root:

\[ x_1 = \frac{{-13.5 + \sqrt{43.05}}}{2 \cdot 2} \]

Using the negative square root for the second root:

\[ x_2 = \frac{{-13.5 - \sqrt{43.05}}}{2 \cdot 2} \]

After calculating the roots, we round them to the nearest thousandths:

\[ x_1 \approx -1.735 \] \[ x_2 \approx -5.015 \]

Final Answer