To solve the quadratic equation \(2x^2 - 8x + 6 = 0\), we will use the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation in the standard form \(ax^2 + bx + c = 0\). ### Step 1: Identify the coefficients From the equation \(2x^2 - 8x + 6 = 0\), we can identify: - \(a = 2\) - \(b = -8\) - \(c = 6\) ### Step 2: Calculate the discriminant The discriminant \(D\) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values of \(a\), \(b\), and \(c\): \[ D = (-8)^2 - 4 \cdot 2 \cdot 6 \] \[ D = 64 - 48 \] \[ D = 16 \] ### Step 3: Apply the quadratic formula Now that we have the discriminant, we can substitute \(a\), \(b\), and \(D\) into the quadratic formula: \[ x = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 2} \] \[ x = \frac{8 \pm 4}{4} \] ### Step 4: Calculate the two possible solutions Now we will calculate the two possible values for \(x\): 1. **First solution**: \[ x_1 = \frac{8 + 4}{4} = \frac{12}{4} = 3 \] 2. **Second solution**: \[ x_2 = \frac{8 - 4}{4} = \frac{4}{4} = 1 \] ### Final Solutions Thus, the solutions to the quadratic equation \(2x^2 - 8x + 6 = 0\) are: - \(x = 3\) - \(x = 1\) ### Correct Option The correct option is **A. \(x = 1\) and \(x = 3\)**. ### Explanation of Other Options - **Option B: \(x = 2\) and \(x = 3\)**: This option is incorrect because \(x = 2\) is not a solution derived from our calculations. - **Option C: \(x = 1\) and \(x = 4\)**: This option is incorrect because \(x = 4\) is not a solution; we found \(x = 3\) instead. - **Option D: \(x = 3\) and \(x = 2\)**: This option is incorrect because, while \(x = 3\) is correct, \(x = 2\) is not a solution. ### Common Pitfalls - **Miscalculating the discriminant**: Always double-check your calculations for \(D\). - **Forgetting to apply the \(\pm\) in the quadratic formula**: This is crucial as it leads to two distinct solutions. - **Confusing the order of solutions**: The order of solutions does not matter, but ensure both solutions are correct. ### Revision Summary - Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find solutions. - Calculate the discriminant \(D = b^2 - 4ac\) to determine the nature of the roots. - Substitute values carefully and remember to consider both \(+\) and \(-\) in the formula. - Verify your solutions by substituting them back into the original equation.