Determine the value of the discriminant. Use the discriminant to determine the most efficient method to solve the quadratic equation, and then solve. \[ 9 x^{2}-6 x+1=0 \]
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To find the value of the discriminant for the quadratic equation \( 9x^2 - 6x + 1 = 0 \), we use the formula: \[ D = b^2 - 4ac \] Here, \( a = 9 \), \( b = -6 \), and \( c = 1 \). Calculating the discriminant: \[ D = (-6)^2 - 4(9)(1) = 36 - 36 = 0 \] Since the discriminant is \( 0 \), this indicates that there is exactly one real solution (a repeated root). In this case, the most efficient method to solve the quadratic equation is by using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values we have: \[ x = \frac{-(-6) \pm \sqrt{0}}{2 \cdot 9} = \frac{6 \pm 0}{18} = \frac{6}{18} = \frac{1}{3} \] Therefore, the solution to the equation \( 9x^2 - 6x + 1 = 0 \) is: \[ x = \frac{1}{3} \]
