Questions: Solve by using the quadratic formula. Express the solution set in exact simplest form. y^2 - 3y - 5 = 0

Solution Steps

To solve the quadratic equation \( y^2 - 3y - 5 = 0 \) using the quadratic formula, identify the coefficients \( a = 1 \), \( b = -3 \), and \( c = -5 \). Then, apply the quadratic formula:

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

This will give the exact solutions for \( y \).

For the quadratic equation \( y^2 - 3y - 5 = 0 \), we identify the coefficients as follows:

• \( a = 1 \)
• \( b = -3 \)
• \( c = -5 \)

The discriminant \( D \) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = (-3)^2 - 4 \cdot 1 \cdot (-5) = 9 + 20 = 29 \]

Using the quadratic formula: \[ y = \frac{-b \pm \sqrt{D}}{2a} \] we substitute \( D = 29 \): \[ y = \frac{-(-3) \pm \sqrt{29}}{2 \cdot 1} = \frac{3 \pm \sqrt{29}}{2} \]

The two solutions are: \[ y_1 = \frac{3 + \sqrt{29}}{2} \quad \text{and} \quad y_2 = \frac{3 - \sqrt{29}}{2} \] Calculating the approximate values: \[ y_1 \approx 4.1926 \quad \text{and} \quad y_2 \approx -1.1926 \]

Final Answer