Find the roots of cach equation. 2y^2-y-5=0

To find the roots of the quadratic equation \(2y^2 - y - 5 = 0\), we can use the quadratic formula:<br /><br />\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br /><br />Here, \(a = 2\), \(b = -1\), and \(c = -5\).<br /><br />First, calculate the discriminant (\(\Delta\)):<br /><br />\[ \Delta = b^2 - 4ac \]<br />\[ \Delta = (-1)^2 - 4(2)(-5) \]<br />\[ \Delta = 1 + 40 \]<br />\[ \Delta = 41 \]<br /><br />Now, substitute \(\Delta\) back into the quadratic formula:<br /><br />\[ y = \frac{-(-1) \pm \sqrt{41}}{2(2)} \]<br />\[ y = \frac{1 \pm \sqrt{41}}{4} \]<br /><br />So, the roots of the equation \(2y^2 - y - 5 = 0\) are:<br /><br />\[ y = \frac{1 + \sqrt{41}}{4} \quad \text{and} \quad y = \frac{1 - \sqrt{41}}{4} \]