91Ó°ÊÓ

For method A, we will perform the following steps: 1. Write the equation in the standard quadratic form 2. Make the leading coefficient (coefficient of the $$x^2$$ term) equal to 1 3. Add and subtract a constant within the parentheses to complete the square 4. Factor the perfect square trinomial 5. Solve for x using square root property Step 1: Write the equation in the standard quadratic form The given equation is already in the standard quadratic form: $$ 9x^2 - 6x + 1 = 0 $$ Step 2: Make the leading coefficient equal to 1 Divide the entire equation by 9: $$ x^2 - \frac{2}{3}x + \frac{1}{9} = 0 $$ Step 3: Add and subtract the constant within parentheses to complete the square Add and subtract $$\left(\frac{\frac{2}{3}}{2}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}$$ inside the parentheses. $$ x^2 - \frac{2}{3}x + \frac{1}{9} - \frac{1}{9} = -\frac{1}{9} $$ Step 4: Factor the perfect square trinomial Factor the left side of the equation: $$ (x-\frac{1}{3})^2 = -\frac{1}{9} $$ Step 5: Solve for x using square root property Apply the square root property to both sides: $$ x - \frac{1}{3} = \pm \sqrt{-\frac{1}{9}} $$ Then, isolate x: $$ x = \frac{1}{3} \pm \sqrt{-\frac{1}{9}} $$ Hence, the two solutions are: $$ x_1 = \frac{1}{3} + \frac{i}{3} $$ and $$ x_2 = \frac{1}{3} - \frac{i}{3} $$

For method B, we will perform the following steps: 1. Identify the coefficients of the quadratic equation 2. Substitute the coefficients into the quadratic formula 3. Simplify to get the solutions of the equation Step 1: Identify the coefficients of the quadratic equation The given equation is of the form $$ax^2+bx+c=0$$, and the coefficients are: $$ a = 9 \\ b = -6 \\ c = 1 $$ Step 2: Substitute the coefficients into the quadratic formula The quadratic formula is given by: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Substitute the values of a, b, and c into the formula: $$ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 9 \cdot 1}}{2 \cdot 9} $$ Step 3: Simplify to get the solutions of the equation Simplify the expression to find the solutions: $$ x = \frac{6 \pm \sqrt{36 - 36}}{18} $$ This simplifies to: $$ x = \frac{6 \pm \sqrt{0}}{18} = \frac{6 \pm 0}{18} = \frac{6}{18} = \frac{1}{3} $$ Since there is no nonzero value inside the square root, the equation has only one real solution: $$ x = \frac{1}{3} $$