Solve equation by using the quadratic formula. List the solutions , separated by commas. 4w^2+3w-5=0 omega =square

To solve the quadratic equation \(4w^2 + 3w - 5 = 0\) using the quadratic formula, we use the formula:<br /><br />\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br /><br />Here, \(a = 4\), \(b = 3\), and \(c = -5\).<br /><br />First, we calculate the discriminant (\(\Delta\)):<br /><br />\[ \Delta = b^2 - 4ac \]<br />\[ \Delta = 3^2 - 4(4)(-5) \]<br />\[ \Delta = 9 + 80 \]<br />\[ \Delta = 89 \]<br /><br />Next, we substitute the values into the quadratic formula:<br /><br />\[ w = \frac{-3 \pm \sqrt{89}}{2 \cdot 4} \]<br />\[ w = \frac{-3 \pm \sqrt{89}}{8} \]<br /><br />So, the solutions are:<br /><br />\[ w = \frac{-3 + \sqrt{89}}{8} \quad \text{and} \quad w = \frac{-3 - \sqrt{89}}{8} \]<br /><br />Therefore, the solutions to the equation \(4w^2 + 3w - 5 = 0\) are:<br /><br />\[ w = \frac{-3 + \sqrt{89}}{8}, \frac{-3 - \sqrt{89}}{8} \]