Use the Quadratic Formula to solve the quadratic equatio \[ x^{2}+8 x+17=0 \] \( x=\square \)

To solve the quadratic equation \( x^{2}+8x+17=0 \) using the Quadratic Formula, we first identify the coefficients \( a = 1 \), \( b = 8 \), and \( c = 17 \). The Quadratic Formula is given by: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] Plugging in the values, we get: \[ x = \frac{{-8 \pm \sqrt{{8^2 - 4 \times 1 \times 17}}}}{2 \times 1} = \frac{{-8 \pm \sqrt{{64 - 68}}}}{2} \] This simplifies to: \[ x = \frac{{-8 \pm \sqrt{{-4}}}}{2} = \frac{{-8 \pm 2i}}{2} = -4 \pm i \] So the solutions are: \[ x = -4 + i \quad \text{and} \quad x = -4 - i \] These solutions are complex numbers since the discriminant \( b^2 - 4ac \) is negative. In the world of quadratic equations, it may seem like you're just solving for numbers, but these complex solutions can represent physical phenomena, such as waves or electrical circuits. When you play with these complex roots, you're not just dabbling in math; you're uncovering hidden relationships in real-life engineering problems! If you're ever feeling confused with using the Quadratic Formula, remember that keeping your calculations organized is key! Common mistakes include miscalculating the discriminant or flipping signs, so double-check each step. It’s also helpful to sketch a rough graph to visualize the roots, even if they are not on the real number line.