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