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

To solve the quadratic equation \( 31 w^{2}-40 w+16=6 w^{2} \) using the quadratic formula, we first need to rearrange the equation into standard form \( ax^2 + bx + c = 0 \). ### Step 1: Rearranging the Equation Subtract \( 6w^2 \) from both sides: \[ 31w^2 - 6w^2 - 40w + 16 = 0 \] This simplifies to: \[ 25w^2 - 40w + 16 = 0 \] ### Step 2: Identifying Coefficients Now, we identify the coefficients \( a \), \( b \), and \( c \): - \( a = 25 \) - \( b = -40 \) - \( c = 16 \) ### Step 3: Applying the Quadratic Formula The quadratic formula is given by: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] ### Step 4: Calculating the Discriminant First, we calculate the discriminant \( b^2 - 4ac \): \[ b^2 = (-40)^2 = 1600 \] \[ 4ac = 4 \cdot 25 \cdot 16 = 1600 \] \[ b^2 - 4ac = 1600 - 1600 = 0 \] ### Step 5: Finding the Roots Since the discriminant is \( 0 \), there is one rational solution: \[ w = \frac{-(-40) \pm \sqrt{0}}{2 \cdot 25} \] \[ w = \frac{40}{50} = \frac{4}{5} \] ### Final Answer The solution to the equation \( 31 w^{2}-40 w+16=6 w^{2} \) is: \[ w = \frac{4}{5} \]