Quadratic Formula (Rational Solutions) Score: \( 2 / 10 \) Penatty: none Question Use the quadratic formula to solve. Express your answer in simplest form. \[ 31 w^{2}-40 w+16=6 w^{2} \]

First, let's rearrange the equation to standard form. We start with: \[ 31w^2 - 40w + 16 = 6w^2 \] Subtract \(6w^2\) from both sides: \[ 31w^2 - 6w^2 - 40w + 16 = 0 \] This simplifies to: \[ 25w^2 - 40w + 16 = 0 \] Now we can use the quadratic formula, which is given by: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 25\), \(b = -40\), and \(c = 16\). First, calculate the discriminant (\(b^2 - 4ac\)): \[ b^2 = (-40)^2 = 1600 \] \[ 4ac = 4 \cdot 25 \cdot 16 = 1600 \] Now plug these into the discriminant: \[ b^2 - 4ac = 1600 - 1600 = 0 \] Since the discriminant is 0, this means there is one real solution. Now, substitute back into the quadratic formula: \[ w = \frac{-(-40) \pm \sqrt{0}}{2 \cdot 25} = \frac{40 \pm 0}{50} = \frac{40}{50} = \frac{4}{5} \] So, the quadratic equation has a single rational solution: \[ \boxed{\frac{4}{5}} \]