Find all solutions by factoring. \( 3 w^{2}-22 w=16 \)

To solve the equation \( 3w^{2} - 22w - 16 = 0 \) by factoring, we start by rearranging it into standard form. This gives us: \[ 3w^{2} - 22w - 16 = 0 \] Now, we find two numbers that multiply to \( 3 \cdot -16 = -48 \) and add up to \(-22\). The numbers \(-24\) and \(2\) work since \(-24 \times 2 = -48\) and \(-24 + 2 = -22\). Next, we can rewrite the middle term: \[ 3w^{2} - 24w + 2w - 16 = 0 \] Now, we group the terms: \[ (3w^{2} - 24w) + (2w - 16) = 0 \] Factor out common terms from each group: \[ 3w(w - 8) + 2(w - 8) = 0 \] Now, factor out \((w - 8)\): \[ (3w + 2)(w - 8) = 0 \] Setting each factor equal to zero gives: 1. \( 3w + 2 = 0 \) → \( w = -\frac{2}{3} \) 2. \( w - 8 = 0 \) → \( w = 8 \) Thus, the solutions to the equation are \( w = -\frac{2}{3} \) and \( w = 8 \). Let’s break it down a little more with some twists! Who knew solving quadratic equations could be a treasure hunt? Picture yourself on a quest for the elusive “w.” Just remember – hunt for pairs that give you the magic number when multiplied and the right vibe when added! And hey, if factoring feels like deciphering a treasure map, don’t hesitate to use the quadratic formula as your backup compass!