To find the solutions to the quadratic equation \(2x^2 - 5x = 6\), we first rewrite it in standard form:
\[ 2x^2 - 5x - 6 = 0 \]
Next, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 2\), \(b = -5\), and \(c = -6\).
We'll first calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = (-5)^2 - 4(2)(-6) \] \[ = 25 + 48 = 73 \]
Now, substituting the values into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{73}}{2(2)} \] \[ = \frac{5 \pm \sqrt{73}}{4} \]
Now we will calculate the two potential solutions:
- \(x_1 = \frac{5 + \sqrt{73}}{4}\)
- \(x_2 = \frac{5 - \sqrt{73}}{4}\)
We will approximate \(\sqrt{73}\):
\(\sqrt{73} \approx 8.544\)
Calculating \(x_1\):
\[ x_1 = \frac{5 + 8.544}{4} \approx \frac{13.544}{4} \approx 3.386 \]
Calculating \(x_2\):
\[ x_2 = \frac{5 - 8.544}{4} \approx \frac{-3.544}{4} \approx -0.886 \]
Thus, the solutions to the equation \(2x^2 - 5x - 6 = 0\) approximate to:
- \(x_1 \approx 3.39\) (rounded to the nearest hundredth)
- \(x_2 \approx -0.89\) (rounded to the nearest hundredth)
So, the solutions are approximately \(3.39\) and \(-0.89\).