To solve the equation \(2x^2 - 5x = 6\), we first rearrange it into standard quadratic form. We do this by subtracting 6 from both sides:
\[ 2x^2 - 5x - 6 = 0 \]
Next, we can use the quadratic formula to find the roots of the equation. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \(a = 2\), \(b = -5\), and \(c = -6\).
We start by calculating the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = (-5)^2 - 4(2)(-6) \] \[ = 25 + 48 \] \[ = 73 \]
Now we can apply the formula:
\[ x = \frac{-(-5) \pm \sqrt{73}}{2(2)} \] \[ = \frac{5 \pm \sqrt{73}}{4} \]
Calculating \(\sqrt{73}\) gives approximately \(8.54\). Now, substitute this value back into the formula:
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For the positive root: \[ x = \frac{5 + 8.54}{4} = \frac{13.54}{4} \approx 3.385 \]
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For the negative root: \[ x = \frac{5 - 8.54}{4} = \frac{-3.54}{4} \approx -0.885 \]
Rounding to the nearest hundredth, we have:
- The positive solution is \(x \approx 3.39\).
- The negative solution is \(x \approx -0.89\).
Among the options provided, the correct solution rounded to the nearest hundredth is:
\(x = -0.89\).