["# Solving the Quadratic Equation ( -4.9t^2 + 20t + 50 = 0 ): A Complete Guide", "Quadratic equations are fundamental in algebra and appear frequently in sciences, engineering, and economics. One such equation is the quadratic formula application presented here:
\n[ -4.9t^2 + 20t + 50 = 0 ]", "This article provides a thorough explanation on how to solve this equation step-by-step, its real-world applications, and why understanding quadratic equations matters.", "---", "## Understanding the Quadratic Equation", "The general form of a quadratic equation is:
\n[ at^2 + bt + c = 0 ]
\nwhere ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ).", "For the equation:
\n[ -4.9t^2 + 20t + 50 = 0 ]
\nwe identify:
\n- ( a = -4.9 )
\n- ( b = 20 )
\n- ( c = 50 )", "---", "## Step 1: The Quadratic Formula", "The quadratic formula solves for ( t ):
\n[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]", "Applying this to our values:", "[
\nt = \frac{-(20) \pm \sqrt{(20)^2 - 4(-4.9)(50)}}{2(-4.9)}
\n]", "---", "## Step 2: Compute the Discriminant", "First calculate the discriminant ( D ):
\n[
\nD = b^2 - 4ac = 20^2 - 4(-4.9)(50) = 400 + 980 = 1380
\n]", "Since ( D = 1380 > 0 ), there are two distinct real solutions.", "---", "## Step 3: Calculate the Square Root of the Discriminant", "[
\n\sqrt{D} = \sqrt{1380} \approx 37.15 \quad \ ext{(calculated to 2 decimal places)}
\n]", "---", "## Step 4: Substitute Values into the Formula", "[
\nt = \frac{-20 \pm 37.15}{2(-4.9)} = \frac{-20 \pm 37.15}{-9.8}
\n]", "Now compute both roots:", "- ( t_1 = \frac{-20 + 37.15}{-9.8} = \frac{17.15}{-9.8} \approx -1.75 )
\n- ( t_2 = \frac{-20 - 37.15}{-9.8} = \frac{-57.15}{-9.8} \approx 5.84 )", "---", "## Final Solutions", "The two real solutions are approximately:
\n[
\n\boxed{t \approx -1.75} \quad \ ext{and} \quad \boxed{t \approx 5.84}
\n]", "---", "## Real-World Applications of This Equation", "Quadratic equations like ( -4.9t^2 + 20t + 50 = 0 ) arise in numerous practical contexts:", "- Projectile Motion: Modeling the trajectory of an object launched into the air, where ( -4.9 ) corresponds to gravity’s deceleration (in m/s²), ( 20 ) to initial velocity components, and ( 50 ) to starting height.
\n- Economics: Calculating revenue or profit models with nonlinear behavior.
\n- Engineering: Analyzing spring dynamics, electrical circuits, or heat transfer situations involving quadratic relationships.", "---", "## Why Learning Quadratic Equations Matters", "Mastering quadratic equations strengthens analytical thinking and problem-solving skills critical in STEM fields. They bridge algebra with real-world modeling, enabling accurate predictions and informed decisions.", "---", "## Summary", "Solving ( -4.9t^2 + 20t + 50 = 0 ) involves:
\n1. Identifying coefficients ( a, b, c )
\n2. Calculating discriminant
\n3. Applying the quadratic formula
\n4. Simplifying to get approximate real solutions
\n5. Interpreting results in context", "This equation exemplifies how math translates into powerful tools for understanding and predicting physical and economic phenomena.", "---", "### FAQ: Common Questions About Quadratic Equations", "How do I recognize if a quadratic exists?
\nOnly equations with ( at^2 ) (nonzero) are quadratics.", "What if the discriminant is negative?
\nNo real solutions—roots are complex.", "How are real-world motions linked to quadratics?
\nProjectile path equations follow ( y = at^2 + bt + c ), modeling elevation over time.", "---", "### Related Reading
\n- How to Graph Quadratic Functions
\n- Applications of Quadratics in Physics
\n- Quadratic Equations: Solving by Factoring and Completing the Square", "---", "Keywords: quadratic equation, solve (-4.9t^2 + 20t + 50 = 0), real roots, discriminant, projectile motion, algebra tutorial, quadratic applications", "---", "If you're studying quadratics or seeking solutions to similar equations, understanding the step-by-step method here will empower your mathematical toolkit! For further practice, try solving other quadratic equations with varying coefficients."]
