Here, \( a = 1 \), \( b = -4 \), \( c = -21 \).

["Understanding Quadratic Equations: A Closer Look at ( a = 1 ), ( b = -4 ), ( c = -21 )", "When studying quadratic equations, selecting specific values for the coefficients ( a ), ( b ), and ( c ) helps deepen your understanding of their behavior and solutions. In this article, we explore the quadratic equation defined by ( a = 1 ), ( b = -4 ), and ( c = -21 ), how to solve it, and what it reveals about parabolas and their roots.", "---", "### The General Form of a Quadratic Equation", "A quadratic equation takes the standard form:", "[\nax^2 + bx + c = 0\n]", "For our example, substituting the given values:", "[\nx^2 - 4x - 21 = 0\n]", "This equation is particularly valuable because the coefficients are simple integers, making it easy to analyze and solve using multiple methods like factoring, completing the square, or the quadratic formula.", "---", "### Step-by-Step Solution: Factoring the Equation", "With ( a = 1 ), factoring is often straightforward. We aim to express the quadratic as a product of two binomials:", "[\nx^2 - 4x - 21 = (x + m)(x + n)\n]", "We need to find two numbers ( m ) and ( n ) such that:", "- ( m \cdot n = -21 ) (product of ( c ))\n- ( m + n = -4 ) (sum of ( b ))", "After testing possible pairs, ( 3 ) and ( -7 ) satisfy both conditions:", "[\n3 \cdot (-7) = -21 \quad \ ext{and} \quad 3 + (-7) = -4\n]", "So, the equation factors as:", "[\n(x + 3)(x - 7) = 0\n]", "---", "### Solving for ( x ): Finding the Roots", "Set each factor equal to zero:", "[\nx + 3 = 0 \quad \Rightarrow \quad x = -3\n]\n[\nx - 7 = 0 \quad \Rightarrow \quad x = 7\n]", "Thus, the solutions are:", "[\nx = -3 \quad \ ext{and} \quad x = 7\n]", "These roots represent the points where the parabola crosses the ( x )-axis.", "---", "### Analyzing the Parabola", "With ( a = 1 ) (positive), the parabola opens upwards. The vertex lies midway between the two roots:", "[\n\ ext{Vertex } x = \frac{-b}{2a} = \frac{4}{2} = 2\n]", "Substituting ( x = 2 ) into the equation:", "[\ny = (2)^2 - 4(2) - 21 = 4 - 8 - 21 = -25\n]", "So, the vertex is at ( (2, -25) ), confirming the parabola’s minimum point.", "---", "### Real-World Applications and Why This Equation Matters", "Quadratic equations like ( x^2 - 4x - 21 = 0 ) commonly model real-world scenarios such as projectile motion, profit optimization, and geometry problems. Understanding how to solve for ( x ) helps solve practical problems efficiently.", "---", "### Tips for Quick Problem Solving", "- Check discriminant (( D = b^2 - 4ac )) to determine root nature:\n Here, ( D = (-4)^2 - 4(1)(-21) = 16 + 84 = 100 > 0 ), confirming two distinct real roots.", "- Use factoring when coefficients are simple integers for faster solutions.\n- Use the quadratic formula if factoring is difficult:\n [\n x = \frac{4 \pm \sqrt{100}}{2} = \frac{4 \pm 10}{2}\n ]", "---", "### Summary", "Working with the equation ( x^2 - 4x - 21 = 0 ) provides clear insight into how coefficients shape a quadratic function. With ( a = 1 ), ( b = -4 ), and ( c = -21 ), factoring reveals roots at ( x = -3 ) and ( x = 7 ), the parabola opens upward, and the vertex lies at ( (2, -25) ). Mastering such equations strengthens foundational algebra skills useful across science, engineering, and economics.", "---", "Keywords: quadratic equation, solve ( x^2 - 4x - 21 = 0 ), factoring quadratics, roots of a quadratic, parabola analysis, discriminant, real roots, algebra practice."]