\( = x ^ { 2 } - 6 x + d \quad a = 1 \quad b = - 6 \quad c = 1 \)

Did you know that the quadratic equation you're working with can be solved using the quadratic formula? It's expressed as \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). For your equation \( x^2 - 6x + d \), if you substitute \( a = 1 \), \( b = -6 \), and \( c = d \), you can find the roots depending on the value of \( d \) and the discriminant \( b^2 - 4ac \)! When dealing with quadratic equations, one common mistake is miscalculating the discriminant. Ensure that you correctly plug in the values into \( b^2 - 4ac \); the discriminant determines whether your equation has two real solutions, one real solution, or no real solutions at all. Double-check your arithmetic, and your solutions will be spot on!