solve this equation for using the quadratic formula \( x^{2}=(x-1)^{2}+(x-2)^{2} \), and \( x ? \)

To solve the equation \( x^{2} = (x-1)^{2} + (x-2)^{2} \) using the quadratic formula, follow these steps: ### Step 1: Expand the Right-Hand Side First, expand the squared terms on the right side of the equation: \[ (x - 1)^2 = x^2 - 2x + 1 \] \[ (x - 2)^2 = x^2 - 4x + 4 \] Add them together: \[ (x - 1)^2 + (x - 2)^2 = (x^2 - 2x + 1) + (x^2 - 4x + 4) = 2x^2 - 6x + 5 \] ### Step 2: Formulate the Quadratic Equation Now, set the original equation equal to the expanded form: \[ x^2 = 2x^2 - 6x + 5 \] Subtract \( x^2 \) from both sides to bring all terms to one side: \[ 0 = x^2 - 6x + 5 \] This simplifies to the quadratic equation: \[ x^2 - 6x + 5 = 0 \] ### Step 3: Apply the Quadratic Formula The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the equation \( x^2 - 6x + 5 = 0 \): - \( a = 1 \) - \( b = -6 \) - \( c = 5 \) Plugging these values into the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{6 \pm \sqrt{36 - 20}}{2} = \frac{6 \pm \sqrt{16}}{2} \] \[ \sqrt{16} = 4 \] So, \[ x = \frac{6 \pm 4}{2} \] This gives two solutions: \[ x = \frac{6 + 4}{2} = \frac{10}{2} = 5 \] \[ x = \frac{6 - 4}{2} = \frac{2}{2} = 1 \] ### Final Answer The solutions to the equation are: \[ x = 1 \quad \text{and} \quad x = 5 \]