3. Solve for the roots of the function below: \[ y=3 x^{2}-10 x-8 \]

To find the roots of the quadratic function: \[ y = 3x^{2} - 10x - 8 \] we need to determine the values of \( x \) for which \( y = 0 \). This means solving the equation: \[ 3x^{2} - 10x - 8 = 0 \] ### Step 1: Identify the Coefficients The given quadratic equation is in the standard form \( ax^{2} + bx + c = 0 \), where: - \( a = 3 \) - \( b = -10 \) - \( c = -8 \) ### Step 2: Use the Quadratic Formula The quadratic formula to find the roots of the equation \( ax^{2} + bx + c = 0 \) is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] Plugging in the values of \( a \), \( b \), and \( c \): \[ x = \frac{-(-10) \pm \sqrt{(-10)^{2} - 4 \cdot 3 \cdot (-8)}}{2 \cdot 3} \] ### Step 3: Simplify Inside the Square Root Calculate the discriminant (\( \Delta \)): \[ \Delta = b^{2} - 4ac = (-10)^{2} - 4 \cdot 3 \cdot (-8) = 100 + 96 = 196 \] Since the discriminant is positive, there are two real roots. ### Step 4: Calculate the Roots Now, substitute the discriminant back into the formula: \[ x = \frac{10 \pm \sqrt{196}}{6} = \frac{10 \pm 14}{6} \] This gives two possible solutions: 1. **First Root (\(+\) sign):** \[ x = \frac{10 + 14}{6} = \frac{24}{6} = 4 \] 2. **Second Root (\(-\) sign):** \[ x = \frac{10 - 14}{6} = \frac{-4}{6} = -\frac{2}{3} \] ### Step 5: Conclusion The roots of the quadratic function \( y = 3x^{2} - 10x - 8 \) are: \[ x = 4 \quad \text{and} \quad x = -\frac{2}{3} \] These are the values of \( x \) where the function crosses the x-axis.