Questions: Find all real solutions of the equation. (6 w-5)^2-2=0 w=

Solution Steps

To solve the equation \((6w - 5)^2 - 2 = 0\), we first isolate the squared term by adding 2 to both sides, resulting in \((6w - 5)^2 = 2\). Next, we take the square root of both sides to solve for \(6w - 5\), which gives us two possible equations: \(6w - 5 = \sqrt{2}\) and \(6w - 5 = -\sqrt{2}\). We then solve each equation for \(w\) by adding 5 to both sides and dividing by 6.

Start with the equation \((6w - 5)^2 - 2 = 0\). Add 2 to both sides to isolate the squared term: \[ (6w - 5)^2 = 2 \]

Take the square root of both sides to solve for \(6w - 5\): \[ 6w - 5 = \pm \sqrt{2} \]

Solve the two resulting equations for \(w\):

1. \(6w - 5 = \sqrt{2}\) \[ 6w = \sqrt{2} + 5 \] \[ w = \frac{\sqrt{2} + 5}{6} \]

2. \(6w - 5 = -\sqrt{2}\) \[ 6w = -\sqrt{2} + 5 \] \[ w = \frac{-\sqrt{2} + 5}{6} \]

Final Answer