Questions: Solve by using the quadratic formula. (Enter your ans w=1/w^2 - 4/w + 3 = 0

Solution Steps

To solve the quadratic equation using the quadratic formula, we need to first rewrite the equation in the standard form \(ax^2 + bx + c = 0\). Then, we can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.

Given the equation: \[ w^3 - 4w^2 + 3w = 0 \] We can factor out \(w\): \[ w(w^2 - 4w + 3) = 0 \] This gives us two equations to solve: \[ w = 0 \] and \[ w^2 - 4w + 3 = 0 \]

For the quadratic equation \(w^2 - 4w + 3 = 0\), we use the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = -4\), and \(c = 3\).

The discriminant \(\Delta\) is: \[ \Delta = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 3 = 16 - 12 = 4 \]

Using the quadratic formula: \[ w = \frac{-(-4) \pm \sqrt{4}}{2 \cdot 1} = \frac{4 \pm 2}{2} \] This gives us two roots: \[ w_1 = \frac{4 + 2}{2} = 3 \] \[ w_2 = \frac{4 - 2}{2} = 1 \]

Final Answer