t = rac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot 2}}{2 \cdot 9} = rac{10 \pm \sqrt{100 - 72}}{18} = rac{10 \pm \sqrt{28}}{18}.
["# Simplifying the Quadratic Formula Result: A Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, and understanding the general solution using the quadratic formula is essential. One common scenario involves expressions like:", "$$\nt = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}\n$$", "This equation arises when applying the quadratic formula — often denoted as:", "$$\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "In this case, identifying coefficients:\n- ( a = 9 )\n- ( b = -10 )\n- ( c = 2 )", "### Step-by-Step Simplification", "Start by computing the discriminant ( D = b^2 - 4ac ):", "$$\nD = (-10)^2 - 4 \cdot 9 \cdot 2 = 100 - 72 = 28\n$$", "Now substitute back into the quadratic formula:", "$$\nt = \frac{10 \pm \sqrt{28}}{18}\n$$", "Since ( \sqrt{28} ) can be simplified:", "$$\n\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}\n$$", "Thus, the expression becomes:", "$$\nt = \frac{10 \pm 2\sqrt{7}}{18}\n$$", "To simplify further, factor numerator and denominator:", "$$\nt = \frac{2(5 \pm \sqrt{7})}{18} = \frac{5 \pm \sqrt{7}}{9}\n$$", "### Final Simplified Result", "$$\nt = \frac{10 \pm \sqrt{28}}{18} = \frac{5 \pm \sqrt{7}}{9}\n$$", "### Why This Simplification Matters", "- Clearer interpretation: Expressed in simplest radical form, making it easier to evaluate numerically or use in further calculations.\n- Reduced numerical error: Working with smaller numbers and simplified radicals helps when computing approximate values.\n- Increased clarity: Simplified expressions are preferred in mathematical communication and academic work.", "### Practical Example", "To approximate ( t ), use the simplified form:", "$$\nt = \frac{5 \pm \sqrt{7}}{9} \approx \frac{5 \pm 2.6458}{9}\n$$", "So the approximate values are:", "- ( t \approx \frac{7.6458}{9} \approx 0.849 )\n- ( t \approx \frac{2.3542}{9} \approx 0.262 )", "---", "This breakdown shows how complex-looking quadratic expressions simplify neatly, enhancing both understanding and accuracy. Whether for studying math, programming algorithms, or engineering applications, mastering these steps leads to clearer problem solving and reliable results."]
