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

To solve the equation using the **quadratic formula**, follow these steps: \[ x^{2} = (x - 1)^{2} + (x - 2)^{2} \] ### Step 1: Expand the Right Side First, expand the squared terms on the right side of the equation: \[ (x - 1)^2 = x^2 - 2x + 1 \] \[ (x - 2)^2 = x^2 - 4x + 4 \] So, the equation becomes: \[ x^2 = (x^2 - 2x + 1) + (x^2 - 4x + 4) \] ### Step 2: Combine Like Terms Combine the terms on the right side: \[ x^2 = 2x^2 - 6x + 5 \] ### Step 3: Rearrange the Equation Move all terms to one side to set the equation to zero: \[ x^2 - 2x^2 + 6x - 5 = 0 \] \[ -x^2 + 6x - 5 = 0 \] Multiply through by -1 to make the coefficient of \( x^2 \) positive: \[ x^2 - 6x + 5 = 0 \] ### Step 4: Apply the Quadratic Formula The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the equation \( x^2 - 6x + 5 = 0 \): - \( a = 1 \) - \( b = -6 \) - \( c = 5 \) Plugging these values into the formula: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \] \[ x = \frac{6 \pm \sqrt{36 - 20}}{2} \] \[ x = \frac{6 \pm \sqrt{16}}{2} \] \[ x = \frac{6 \pm 4}{2} \] ### Step 5: Solve for \( x \) Calculate both possible solutions: 1. **Positive Root:** \[ x = \frac{6 + 4}{2} = \frac{10}{2} = 5 \] 2. **Negative Root:** \[ x = \frac{6 - 4}{2} = \frac{2}{2} = 1 \] ### Step 6: Verify the Solutions **For \( x = 1 \):** \[ 1^2 = (1 - 1)^2 + (1 - 2)^2 \\ 1 = 0 + 1 \\ 1 = 1 \quad \checkmark \] **For \( x = 5 \):** \[ 5^2 = (5 - 1)^2 + (5 - 2)^2 \\ 25 = 16 + 9 \\ 25 = 25 \quad \checkmark \] Both solutions satisfy the original equation. ### **Final Answer:** The solutions are \( x = 1 \) and \( x = 5 \).