For the function \( h(x) = x^2 - 4x + c \), we have \( a = 1 \), \( b = -4 \), and \( c = c \). Set the discriminant to zero:
["Understanding the Quadratic Function ( h(x) = x^2 - 4x + c ) and Setting Its Discriminant to Zero", "Quadratic functions are fundamental in algebra, essential in both theoretical math and real-world applications like physics, engineering, and economics. One of the key properties of a quadratic equation ( ax^2 + bx + c = 0 ) is its discriminant, which determines the nature of its roots. In this article, we’ll explore the specific case of the function ( h(x) = x^2 - 4x + c ), where ( a = 1 ), ( b = -4 ), and ( c = c ), and learn how setting the discriminant to zero helps identify unique solutions.", "---", "### What Is the Discriminant?", "For any quadratic equation in standard form ( ax^2 + bx + c = 0 ), the discriminant ( D ) is given by:", "[\nD = b^2 - 4ac\n]", "The discriminant reveals the nature of the roots:\n- If ( D > 0 ): two distinct real roots\n- If ( D = 0 ): one real root (a repeated or double root)\n- If ( D < 0 ): two complex conjugate roots", "Setting the discriminant to zero—( D = 0 )—is especially powerful because it guarantees a single real solution, a critical point that makes kink or tangent to the x-axis in the function’s graph.", "---", "### Analyzing ( h(x) = x^2 - 4x + c )", "Given:\n( a = 1 ),\n( b = -4 ),\n( c = c )", "Plug into the discriminant formula:", "[\nD = (-4)^2 - 4(1)(c) = 16 - 4c\n]", "We want the discriminant to be zero:", "[\n16 - 4c = 0\n]", "Solve for ( c ):", "[\n4c = 16 \quad \Rightarrow \quad c = 4\n]", "---", "### Why Setting ( D = 0 ) Matters", "When ( c = 4 ), the quadratic function becomes:", "[\nh(x) = x^2 - 4x + 4\n]", "This simplifies perfectly to a perfect square trinomial:", "[\nh(x) = (x - 2)^2\n]", "This shows that the graph touches the x-axis at exactly one point: ( x = 2 ), confirming a repeated root. At this point, ( h(2) = 0 ), making ( (2, 0) ) the vertex of the parabola touching its minimum.", "---", "### Practical Applications", "- Finding Tangent Lines: A discriminant of zero ensures tangency, important in optimization and geometry.\n- Modeling Real-World Scenarios: Ensuring unique solutions can represent balanced systems, such as when forces or costs intersect at a single optimal point.\n- Quadratic Completing the Square: Setting ( D = 0 ) confirms a function’s vertex form, simplifying graphing and analysis.", "---", "### Summary", "For the quadratic function ( h(x) = x^2 - 4x + c ),\n( a = 1 ), ( b = -4 ), and setting the discriminant ( D = b^2 - 4ac = 0 ) yields ( c = 4 ).\nThis ensures a unique real root, a repeated solution, and a parabolic graph that touches the x-axis at a single point.", "Understanding when and why to set the discriminant to zero is a crucial skill—transforming abstract equations into meaningful geometric and real-world insights.", "---", "Keywords:\nquadratic function, discriminant, h(x), x² - 4x + c, discriminant zero, repeated root, algebra, graphing quadratics, perfect square trinomial, vertex form", "---", "Call to Action:\nNext time you analyze a quadratic, check the discriminant—especially when looking for unique solutions or tangent behavior. Mastering this will deepen your grasp of quadratic relationships and enhance your problem-solving skills!"]
