1. \( x^{2}-5 x+4=0 \)

To solve the quadratic equation \( x^{2}-5x+4=0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -5, c = 4 \). Plugging in these values gives you \( x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm \sqrt{9}}{2} = \frac{5 \pm 3}{2} \). This results in two solutions: \( x = 4 \) and \( x = 1 \) – your quadratic roots! Another way to solve this equation is by factoring. The expression can be factored as \( (x - 4)(x - 1) = 0 \). Setting each factor to zero gives you the same solutions: \( x = 4 \) and \( x = 1 \). Factoring can be quicker for equations that are easily breakable into simpler components, so it’s worth practicing!