To solve the quadratic equation \( 2x^2 - 9x = 4 \), we first need to rearrange it into standard form:
\[ 2x^2 - 9x - 4 = 0. \]
Next, we can use the quadratic formula to find the solutions, where \( a = 2 \), \( b = -9 \), and \( c = -4 \). The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
Now we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-9)^2 = 81, \]
\[ 4ac = 4 \times 2 \times (-4) = -32, \]
\[ b^2 - 4ac = 81 - (-32) = 81 + 32 = 113. \]
Since the discriminant is positive, there will be two real solutions. Now we find the roots:
\[ x = \frac{-(-9) \pm \sqrt{113}}{2 \times 2} = \frac{9 \pm \sqrt{113}}{4}. \]
Next, we estimate \( \sqrt{113} \). Since \( 10^2 = 100 \) and \( 11^2 = 121 \), we know that \( \sqrt{113} \) is between 10 and 11. A more precise estimate gives approximately \( 10.630\). Thus,
\[ x \approx \frac{9 \pm 10.630}{4}. \]
Calculating the two possible values of \( x \):
- For \( 9 + 10.630 \):
\[ x_1 \approx \frac{19.630}{4} \approx 4.9075 \quad (rounded \ to \ 4.91) \]
- For \( 9 - 10.630 \):
\[ x_2 \approx \frac{-1.630}{4} \approx -0.4075 \quad (rounded \ to \ -0.41) \]
However, we see that we need to report the answers rounded to the nearest hundredth as needed.
Now rounding our results:
- \( x_1 \approx 4.91 \)
- \( x_2 \approx -0.41 \)
Given the provided responses, the one that matches is \( x = 4.91 \).
Therefore, the correct answer is:
x = 4.91.