2y^2-y-6=0

To solve the quadratic equation \(2y^2 - y - 6 = 0\), you can use the quadratic formula:

\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation:

• \(a = 2\)
• \(b = -1\)
• \(c = -6\)

Now, let's calculate the discriminant \(b^2 - 4ac\):

\[ b^2 - 4ac = (-1)^2 - 4(2)(-6) = 1 + 48 = 49 \]

Now, substitute the values back into the quadratic formula:

\[ y = \frac{-(-1) \pm \sqrt{49}}{2(2)} = \frac{1 \pm 7}{4} \]

This gives us two possible solutions:

1. When using the plus sign: \[ y = \frac{1 + 7}{4} = \frac{8}{4} = 2 \]

2. When using the minus sign: \[ y = \frac{1 - 7}{4} = \frac{-6}{4} = -\frac{3}{2} \]

Thus, the solutions to the equation \(2y^2 - y - 6 = 0\) are:

\[ y = 2 \quad \text{and} \quad y = -\frac{3}{2} \]