In a development project, a quadratic function \( h(x) = x^2 - 4x + c \) models the height of a support beam, where \( c \) is a constant. Determine \( c \) such that the function has exactly one real root.

["SEO-Optimized Article: Finding the Value of ( c ) for Exactly One Real Root in the Quadratic Function ( h(x) = x^2 - 4x + c )", "---", "Optimizing Structural Integrity: Finding ( c ) for a Quadratic Height Function with Exactly One Real Root", "In engineering and civil development projects, precise modeling of structural elements is critical. Consider a key scenario where the height of a support beam is modeled by the quadratic function:", "[\nh(x) = x^2 - 4x + c\n]", "where ( c ) is a constant determined based on design constraints. Engineers often face a crucial question: For what value of ( c ) does this support beam height function have exactly one real root? Understanding this ensures optimal beam positioning—critical for stability and material efficiency.", "---", "### Why Does the Number of Real Roots Matter?", "A quadratic equation ( h(x) = x^2 - 4x + c ) can have:", "- Two distinct real roots (when the parabola crosses the x-axis twice),\n- One real root (when the parabola just touches the x-axis),\n- No real roots (when it lies entirely above the x-axis).", "In structural applications, a single real root means the beam touches the base at exactly one point—ideal for minimal material use and maximum load-bearing efficiency.", "---", "### Conditions for Exactly One Real Root", "A quadratic equation ( ax^2 + bx + c = 0 ) has exactly one real root when its discriminant ( D = b^2 - 4ac ) equals zero.", "For our function ( h(x) = x^2 - 4x + c ):\n- ( a = 1 )\n- ( b = -4 )\n- ( c = c ) (a constant variable)", "Compute the discriminant:", "[\nD = (-4)^2 - 4(1)(c) = 16 - 4c\n]", "Set ( D = 0 ) for exactly one real root:", "[\n16 - 4c = 0\n]", "Solve for ( c ):", "[\n4c = 16 \implies c = 4\n]", "---", "### Interpretation and Engineering Insight", "When ( c = 4 ), the height function becomes:", "[\nh(x) = x^2 - 4x + 4 = (x - 2)^2\n]", "This confirms the parabola touches the x-axis at ( x = 2 )—a single point of contact. This minimal contact ensures efficient load transfer to the foundation without redundant support material.", "From an SEO perspective, targeting terms like quadratic function with exactly one real root, discriminant condition for quadratic roots, and support beam structural modeling helps your article rank for technical queries in civil engineering and mathematical optimization.", "---", "### Final Answer", "[\n\boxed{c = 4}\n]", "Using ( c = 4 ) ensures the support beam modeled by ( h(x) = x^2 - 4x + c ) has exactly one real root—achieving optimal structural performance in development projects.", "---", "Keywords:\nquadratic function, real root discriminant, support beam modeling, structural engineering, one real root, ( h(x) = x^2 - 4x + c ), engineering optimization", "Meta Description:\nExplore how the constant ( c ) in ( h(x) = x^2 - 4x + c ) determines a beam’s contact with the foundation. Discover how setting ( c = 4 ) yields exactly one real root—ideal for minimal yet effective structural design.", "---", "For more solutions in quadratic modeling, visit engineering math guides and optimization tutorials tailored for civil development projects."]