For a quadratic equation \( ax^2 + bx + c = 0 \) to have exactly one real root, its discriminant must be zero. The discriminant \(\Delta\) is given by:

["Understanding When a Quadratic Equation Has Exactly One Real Root", "For any quadratic equation of the form\n[ ax^2 + bx + c = 0 ]\nwith ( a <br/>\neq 0 ), the number of real roots depends on the discriminant, a value derived from the coefficients ( a ), ( b ), and ( c ). Specifically, the discriminant is given by:\n[ \Delta = b^2 - 4ac ]", "A key property of the discriminant is that it determines the nature of the roots:\n- If ( \Delta > 0 ): two distinct real roots\n- If ( \Delta = 0 ): exactly one real root (a repeated root)\n- If ( \Delta < 0 ): no real roots (two complex conjugate roots)", "---", "### Why Does the Discriminant Determine the Number of Real Roots?", "The roots of a quadratic equation are found using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{\Delta}}{2a}\n]", "When ( \Delta = 0 ), the square root becomes zero, and the two solutions collapse into a single value:\n[\nx = \frac{-b \pm 0}{2a} = \frac{-b}{2a}\n]\nThis proves there is exactly one real root—a repeated root—meaning the parabola touches the x-axis at exactly one point.", "---", "### How to Find When a Quadratic Has Exactly One Real Root", "To determine if ( ax^2 + bx + c = 0 ) has exactly one real root, follow these steps:", "1. Calculate the discriminant:\n[ \Delta = b^2 - 4ac ]", "2. Check the value of ( \Delta ):\n- If ( \Delta = 0 ): One real root\n- If ( \Delta > 0 ): Two distinct real roots\n- If ( \Delta < 0 ): No real roots", "Only when ( \Delta = 0 ) does the equation have precisely one solution.", "---", "### Example", "Consider the equation:\n[ 2x^2 - 8x + 8 = 0 ]", "Compute the discriminant:\n[\n\Delta = (-8)^2 - 4(2)(8) = 64 - 64 = 0\n]\nSince ( \Delta = 0 ), the equation has exactly one real root:\n[\nx = \frac{8}{2 \cdot 2} = \frac{8}{4} = 2\n]", "---", "### Conclusion", "Understanding the discriminant is essential in algebra—especially when analyzing how a quadratic equation behaves. A discriminant of zero signals the special case where the quadratic has exactly one real root, emphasizing symmetry and touching of the parabola to the x-axis. This concept is foundational in solving equations, graphing quadratics, and applying algebraic principles across science and engineering.", "For more insights on discriminants and quadratic behavior, explore related resources and practice problems—mastery of this concept unlocks deeper mathematical reasoning."]