To solve the quadratic equation \(2x^2 - 9x - 4 = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 2\), \(b = -9\), and \(c = -4\).
First, calculate the discriminant:
\[ b^2 - 4ac = (-9)^2 - 4(2)(-4) = 81 + 32 = 113 \]
Now, substitute the values into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{113}}{2(2)} = \frac{9 \pm \sqrt{113}}{4} \]
Next, calculate the square root of 113:
\(\sqrt{113} \approx 10.63014581273465\)
Now, calculate the two possible values for \(x\):
-
\(x_1 = \frac{9 + 10.63014581273465}{4} \approx \frac{19.63014581273465}{4} \approx 4.9075364531836625 \approx 4.91\) (rounded to the nearest hundredth)
-
\(x_2 = \frac{9 - 10.63014581273465}{4} \approx \frac{-1.63014581273465}{4} \approx -0.4075364531836625 \approx -0.41\) (rounded to the nearest hundredth)
Now, from the provided options:
- \(x = 4.91\)
- \(x = -0.91\) - incorrect
- \(x = 4.77\)
- \(x = -0.77\)
The solution \(x = 4.91\) is a match.
Thus, the correct answer is \(x = 4.91\).