\[ t = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-19,995)}}{2 \times 1} \]

["# Solving the Quadratic Equation:\n[ t = \frac{-3 \pm \sqrt{3^2 - 4 \ imes 1 \ imes (-19{,}995)}}{2 \ imes 1] ]", "When faced with a quadratic equation in standard form ( at^2 + bt + c = 0 ), solving for ( t ) is a fundamental algebraic skill. In this article, we explore one specific quadratic equation:", "[\nt = \frac{-3 \pm \sqrt{3^2 - 4 \ imes 1 \ imes (-19{,}995)}}{2 \ imes 1}\n]", "This equation follows the general quadratic formula:", "[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "### Step 1: Identify coefficients ( a ), ( b ), and ( c )", "From the given expression:\n- ( a = 1 )\n- ( b = -3 )\n- ( c = -19{,}995 )", "### Step 2: Compute the discriminant", "The discriminant ( D ) determines the nature of the roots:", "[\nD = b^2 - 4ac = (-3)^2 - 4 \ imes 1 \ imes (-19{,}995)\n]", "[\nD = 9 + 79{,}980 = 79{,}989\n]", "Since the discriminant is positive, the equation has two distinct real roots.", "### Step 3: Simplify the square root\nThe square root of 79,989 is not a perfect square, but we can approximate or leave it in radical form:", "[\n\sqrt{79{,}989} \approx 282.8 , (\ ext{approximate decimal value})\n]", "### Step 4: Substitute into the quadratic formula", "[\nt = \frac{-3 \pm \sqrt{79{,}989}}{2}\n]", "This yields two solutions:", "[\nt_1 = \frac{-3 + 282.8}{2} = \frac{279.8}{2} = 139.9\n]\n[\nt_2 = \frac{-3 - 282.8}{2} = \frac{-285.8}{2} = -142.9\n]", "### Step 5: Interpret the solutions", "The roots of the equation:\n[\nt = \frac{-3 \pm \sqrt{79{,}989}}{2}\n]\nare approximately:\n- ( t \approx 139.9 )\n- ( t \approx -142.9 )", "These represent the points where the quadratic function crosses the ( t )-axis.", "### Final Thoughts", "Working through quadratic equations step-by-step not only confirms algebraic proficiency but also strengthens problem-solving intuition. Understanding the discriminant helps anticipate root types—here, a large positive discriminant confirms two real, irrational roots.", "Use this formula as a template for solving any quadratic equation efficiently and accurately.", "---", "### Optimized Keywords for SEO:\n- Quadratic equation solution\n- Solve quadratic using discriminant\n- Solve ( t = \frac{-3 \pm \sqrt{79{,}989}}{2} )\n- Real roots of a quadratic\n- Discriminant and real roots\n- Systematic quadratic formula steps", "Whether you're a student mastering algebra or a teacher explaining quadratic formulas, mastering expressions like this equation ensures you confidently handle standard and non-standard quadratic cases."]