To solve the quadratic equation \(2x^2 - 5x = 6\), we first rearrange it into standard form:
\[ 2x^2 - 5x - 6 = 0 \]
Next, we will apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 2\), \(b = -5\), and \(c = -6\).
Calculating \(b^2 - 4ac\):
\[ b^2 = (-5)^2 = 25 \]
\[ 4ac = 4 \cdot 2 \cdot (-6) = -48 \]
So,
\[ b^2 - 4ac = 25 - (-48) = 25 + 48 = 73 \]
Now substituting into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{73}}{2 \cdot 2} \]
\[ x = \frac{5 \pm \sqrt{73}}{4} \]
Calculating \(\sqrt{73}\):
\(\sqrt{73} \approx 8.544\) (rounded to three decimal places)
Now, substituting this value back into the equation for \(x\):
\[ x = \frac{5 \pm 8.544}{4} \]
Calculating the two possible solutions:
- \(x_1 = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386\)
- \(x_2 = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886\)
Rounding \(x_2\) to the nearest hundredth gives:
\[ x_2 \approx -0.89 \]
Thus, the solution to the quadratic equation is:
C) \(x = -0.89\).