1. The problem is to solve the quadratic equation $x^2 - 4x - 42 = 0$ using the quadratic formula. 2. The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. Identify the coefficients from the equation $x^2 - 4x - 42 = 0$: - $a = 1$ - $b = -4$ - $c = -42$ 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-42) = 16 + 168 = 184$$ 5. Substitute the values into the quadratic formula: $$x = \frac{-(-4) \pm \sqrt{184}}{2 \times 1} = \frac{4 \pm \sqrt{184}}{2}$$ 6. Simplify the square root: $$\sqrt{184} = \sqrt{4 \times 46} = 2\sqrt{46}$$ 7. So the solutions are: $$x = \frac{4 \pm 2\sqrt{46}}{2} = 2 \pm \sqrt{46}$$ 8. Therefore, the two roots of the quadratic equation are: $$x_1 = 2 + \sqrt{46}$$ $$x_2 = 2 - \sqrt{46}$$