To determine which of the provided values is a solution to the quadratic equation \(2x^2 - 9 = 4\), we first rewrite the equation:
\[ 2x^2 - 9 - 4 = 0 \quad \Rightarrow \quad 2x^2 - 13 = 0 \]
Next, we can solve for \(x^2\):
\[ 2x^2 = 13 \quad \Rightarrow \quad x^2 = \frac{13}{2} \quad \Rightarrow \quad x^2 = 6.5 \]
Now, we take the square root of both sides to find \(x\):
\[ x = \pm \sqrt{6.5} \]
Calculating \(\sqrt{6.5}\):
\[ \sqrt{6.5} \approx 2.5495 \]
Thus, the solutions are:
\[ x \approx 2.55 \quad \text{and} \quad x \approx -2.55 \]
Now we can compare the approximate values:
- \(2.55\) and \(-2.55\) would round to \(2.55\) and \(-2.55\) respectively which do not match any of the provided options.
Now let's manually check the given options to find which one is close to these values:
- \(x = -0.91\)
- \(x = 4.91\)
- \(x = 4.77\)
- \(x = -0.77\)
None of these values are close to \(-2.55\) or \(2.55\). Let's instead verify each by substituting them back into the original equation \(2x^2 - 9 = 4\).
For \(x = -0.91\): \[ 2(-0.91)^2 - 9 \approx 2(0.8281) - 9 \approx 1.6562 - 9 = -7.3438 \quad \text{(not a solution)} \]
For \(x = 4.91\): \[ 2(4.91)^2 - 9 \approx 2(24.2081) - 9 \approx 48.4162 - 9 = 39.4162 \quad \text{(not a solution)} \]
For \(x = 4.77\): \[ 2(4.77)^2 - 9 \approx 2(22.7529) - 9 \approx 45.5058 - 9 = 36.5058 \quad \text{(not a solution)} \]
For \(x = -0.77\): \[ 2(-0.77)^2 - 9 \approx 2(0.5929) - 9 \approx 1.1858 - 9 = -7.8142 \quad \text{(not a solution)} \]
Since none of the values provided solve the equation, we conclude that none of the provided options is a solution to the equation \(2x^2 - 9 = 4\). If you have further questions or another task, feel free to ask!