A parabola given by \( y = x^2 - 4x + 3 \) intersects the x-axis at which two points?

["### How to Find the x-Intercepts of the Parabola ( y = x^2 - 4x + 3 )", "Understanding where a parabola intersects the x-axis is fundamental in algebra and calculus. These intersection points, also known as roots or x-intercepts, occur where the value of ( y = 0 ). For the quadratic equation ( y = x^2 - 4x + 3 ), finding these points helps in analyzing the shape and behavior of the curve.", "#### Step 1: Set ( y = 0 )\nTo find the x-intercepts, substitute ( y = 0 ):", "[\nx^2 - 4x + 3 = 0\n]", "#### Step 2: Solve the Quadratic Equation\nThis equation can be solved using factoring, completing the square, or the quadratic formula. Here, factoring works efficiently.", "Look for two numbers that multiply to ( 3 ) and add to ( -4 ).\nThose numbers are ( -1 ) and ( -3 ).", "So, factor the quadratic:", "[\n(x - 1)(x - 3) = 0\n]", "#### Step 3: Find the Roots\nSet each factor equal to zero:", "[\nx - 1 = 0 \quad \Rightarrow \quad x = 1\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "#### Step 4: Conclusion\nThe parabola ( y = x^2 - 4x + 3 ) intersects the x-axis at ( x = 1 ) and ( x = 3 ). These points are ( (1, 0) ) and ( (3, 0) ).", "#### Why This Matters\nThese x-intercepts indicate where the parabola crosses the horizontal x-axis and help visualize its symmetry and orientation. Additionally, knowing the roots aids in solving real-world problems such as projectile motion and optimization.", "For further analysis, plotting these points confirms the parabola opens upwards (since the coefficient of ( x^2 ) is positive) and crosses the x-axis at ( x = 1 ) and ( x = 3 ).", "---", "Summary:\nThe parabola ( y = x^2 - 4x + 3 ) intersects the x-axis at the two points:\n[\n\boxed{(1, 0) \ ext{ and } (3, 0)}\n]"]