2x² + 21x - 36 = 0 - United Radiology

["# Solving the Quadratic Equation: 2x² + 21x - 36 = 0", "Solving quadratic equations is a fundamental skill in algebra, often essential for students, educators, and professionals in STEM disciplines. One such equation commonly encountered is 2x² + 21x - 36 = 0. Whether you’re preparing for exams or tackling real-world problems involving parabolas, projectile motion, or optimization, understanding how to solve this equation provides valuable insight. In this article, we walk through step-by-step solutions, interpret the results, and explore how to apply quadratic equations in practical scenarios.", "---", "## Understanding the Quadratic Equation", "The standard form of a quadratic equation is:", "$$ Ax^2 + Bx + C = 0 $$", "For the equation 2x² + 21x - 36 = 0, the coefficients are:", "- A = 2
\n- B = 21
\n- C = -36", "Because ( A <br/>\neq 0 ), the equation remains quadratic.", "---", "## Step-by-Step Solution Using the Quadratic Formula", "The most reliable method for solving any quadratic equation is the quadratic formula:", "$$
\nx = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\n$$", "### Step 1: Compute the Discriminant", "The discriminant, ( D = B^2 - 4AC ), determines the nature of the roots:", "$$
\nD = 21^2 - 4(2)(-36) = 441 + 288 = 729
\n$$", "Since ( D = 729 > 0 ), there are two distinct real solutions.", "### Step 2: Take the Square Root of the Discriminant", "$$
\n\sqrt{729} = 27
\n$$", "### Step 3: Apply the Quadratic Formula", "$$
\nx = \frac{-21 \pm 27}{2 \ imes 2} = \frac{-21 \pm 27}{4}
\n$$", "Now compute both roots:", "- First root:
\n$$
\nx_1 = \frac{-21 + 27}{4} = \frac{6}{4} = \frac{3}{2} = 1.5
\n$$", "- Second root:
\n$$
\nx_2 = \frac{-21 - 27}{4} = \frac{-48}{4} = -12
\n$$", "---", "## Final Solutions", "The solutions to the equation 2x² + 21x - 36 = 0 are:", "$$
\n\boxed{x = \frac{3}{2} \quad \ ext{and} \quad x = -12}
\n$$", "---", "## Application: Graphing the Quadratic Function", "The quadratic equation represents the parabola ( y = 2x² + 21x - 36 ). Knowing the roots allows you to:", "- Plot the x-intercepts
\n- Determine the vertex’s x-coordinate (axis of symmetry) ( x = \frac{-B}{2A} = -\frac{21}{4} = -5.25 )
\n- Identify the direction the parabola opens (upward, since ( A = 2 > 0 ))
\n- Analyze the function’s domain, range, and vertex", "These features are crucial for graphing, optimization, and real-world modeling.", "---", "## Real-World Example: Projectile Motion", "Suppose a ball is thrown upward, and its height in meters after ( x ) seconds is modeled by:", "$$
\nh(x) = -5x² + 21x + 4
\n$$", "To find when the ball hits the ground (height = 0), solve:", "$$
\n-5x² + 21x + 4 = 0
\n$$", "This closely resembles our original equation (with adjusted coefficients), showing how quadratic solutions model physical phenomena like motion under gravity.", "---", "## Summary", "- The equation 2x² + 21x - 36 = 0 has two real solutions: ( x = \frac{3}{2} ) and ( x = -12 )
\n- The discriminant confirms two distinct real roots
\n- This equation can graph parabolas, solve physical problems, and analyze quadratic behavior
\n- Always verify your answer by substituting back into the original equation", "---", "## Further Reading & Practice", "To master quadratic equations:", "- Understand completing the square
\n- Learn to interpret vertex form
\n- Practice solving equations with different coefficients
\n- Explore apps and online tools for interactive graphing", "Mastering quadratics opens the door to algebra’s next big concepts — polynomials, functions, and calculus.", "---", "## Keywords:
\n2x² + 21x - 36 = 0, quadratic equation solution, quadratic formula, parabola graph, real roots, real-world quadratic applications, algebra, high school math, solving quadratic equations, discriminant, vertex form, projectile motion, exponential motion models.", "---", "If you're studying mathematics or preparing for standardized tests, mastering skills like solving 2x² + 21x - 36 = 0 equips you with timeless problem-solving techniques. Use this guide to reinforce your understanding and confidently tackle more complex quadratics ahead."]