3t^2 - 2t + 5 = 20

["Solving the Equation 3t² – 2t + 5 = 20: A Step-by-Step Guide (in Algebra)", "Understanding how to solve quadratic equations is a fundamental skill in algebra, and one common type of problem students encounter is equations of the form 3t² – 2t + 5 = 20. Whether you're preparing for exams or tackling homework, knowing how to simplify and solve this equation can boost your confidence and accuracy. In this article, we’ll walk through the process step-by-step and explore practical problem-solving techniques.", "---", "### Understanding the Equation: 3t² – 2t + 5 = 20", "The equation\n3t² – 2t + 5 = 20\nis a quadratic equation because it contains a term with t², a first-degree term in t, and a constant.", "Our goal is to solve for t by isolating the variable on one side.", "---", "### Step 1: Simplify Both Sides", "Start by moving all terms to one side to set the equation to zero:", "[\n3t² – 2t + 5 – 20 = 0\n]", "Simplify:", "[\n3t² – 2t – 15 = 0\n]", "This is now a standard quadratic equation in the form:\nat² + bt + c = 0, where:\n- a = 3\n- b = –2\n- c = –15", "---", "### Step 2: Applying the Quadratic Formula", "Since factoring may be difficult (or unnecessary), we use the quadratic formula, which is valid for any quadratic equation:", "[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute a, b, and c:", "[\nt = \frac{-(−2) \pm \sqrt{(-2)^2 – 4(3)(–15)}}{2(3)}\n]", "Simplify step by step:", "- ( -(−2) = +2 )\n- ( (-2)^2 = 4 )\n- ( 4 \cdot 3 \cdot (-15) = -180 ), so ( -4ac = -4(3)(-15) = +180 )\n- Discriminant: ( b^2 - 4ac = 4 + 180 = 184 )", "So now:", "[\nt = \frac{2 \pm \sqrt{184}}{6}\n]", "---", "### Step 3: Simplify the Square Root", "Since ( \sqrt{184} ) is not a perfect square, simplify it:", "[\n\sqrt{184} = \sqrt{4 \cdot 46} = 2\sqrt{46}\n]", "Plug back into the formula:", "[\nt = \frac{2 \pm 2\sqrt{46}}{6}\n]", "Factor numerator:", "[\nt = \frac{2(1 \pm \sqrt{46})}{6} = \frac{1 \pm \sqrt{46}}{3}\n]", "---", "### Final Solutions", "The two real solutions are:", "[\nt = \frac{1 + \sqrt{46}}{3} \quad \ ext{and} \quad t = \frac{1 - \sqrt{46}}{3}\n]", "These can be approximated numerically:", "- ( t \approx \frac{1 + 6.78}{3} = \frac{7.78}{3} \approx 2.59 )\n- ( t \approx \frac{1 - 6.78}{3} = \frac{-5.78}{3} \approx -1.93 )", "---", "### Practice: Why Solve This Equation?", "Solving equations like 3t² – 2t + 5 = 20 helps develop algebraic reasoning and prepares learners for advanced math topics, including calculus, physics problems, and engineering calculations. It also builds confidence in handling real-world modeling scenarios where quadratic relationships appear.", "---", "### Tips for Efficiently Solving Quadratics", "- Always simplify the equation first by moving constants to one side.\n- Use the quadratic formula when factoring is not straightforward.\n- Simplify radicals and irrational numbers where possible.\n- Check answers by plugging values back into the original equation.", "---", "### Conclusion", "Solving 3t² – 2t + 5 = 20 leads to a clean application of the quadratic formula, yielding two precise solutions involving a square root. Whether for academic success or real-life problem solving, mastering these steps equips you with a powerful algebraic tool. Keep practicing, and remember: consistent effort turns complex equations into simple solutions!", "---", "Keywords: quadratic equation solver, solve 3t² – 2t + 5 = 20, quadratic formula, algebra problems, solving equations step-by-step, mathematical equations, t quadratic solution, math tutorials, algebra practice.", "---", "Need more help? Explore online calculators, algebra guides, or textbook exercises to reinforce your understanding and tackle harder quadratics."]