Solving the quadratic \( t^2 - 15t + 56 = 0 \) using the quadratic formula:

["# Solving the Quadratic Equation ( t^2 - 15t + 56 = 0 ) Using the Quadratic Formula", "Quadratic equations are foundational in algebra and appear frequently in science, engineering, and economics. One of the most reliable methods to solve any quadratic equation is the quadratic formula. In this article, we’ll explore how to solve the equation ( t^2 - 15t + 56 = 0 ) step-by-step using the quadratic formula, explaining each part to help you understand and apply this powerful technique.", "## What is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation of the form:", "[\nat^2 + bt + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\ne 0 ). The solutions to this equation can be found using the quadratic formula:", "[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "When applied correctly, this formula delivers all real or complex roots of the equation.", "## Applying the Quadratic Formula to ( t^2 - 15t + 56 = 0 )", "### Step 1: Identify coefficients ( a ), ( b ), and ( c )", "For the equation:", "[\nt^2 - 15t + 56 = 0\n]", "we identify:", "- ( a = 1 )\n- ( b = -15 )\n- ( c = 56 )", "### Step 2: Calculate the Discriminant", "The discriminant ( D ) is the part of the formula inside the square root:", "[\nD = b^2 - 4ac\n]", "Substitute the values:", "[\nD = (-15)^2 - 4(1)(56) = 225 - 224 = 1\n]", "Since ( D = 1 > 0 ), there are two distinct real roots.", "### Step 3: Plug into the Quadratic Formula", "[\nt = \frac{-(-15) \pm \sqrt{1}}{2 \cdot 1} = \frac{15 \pm 1}{2}\n]", "This gives two solutions:", "[\nt = \frac{15 + 1}{2} = \frac{16}{2} = 8\n]", "and", "[\nt = \frac{15 - 1}{2} = \frac{14}{2} = 7\n]", "### Step 4: Verify the Solutions", "Check each value in the original equation:", "For ( t = 8 ):\n[\n8^2 - 15 \cdot 8 + 56 = 64 - 120 + 56 = 0\n]", "For ( t = 7 ):\n[\n7^2 - 15 \cdot 7 + 56 = 49 - 105 + 56 = 0\n]", "Both satisfy the equation.", "## Why Use the Quadratic Formula?", "The quadratic formula provides a direct and universal method to solve any quadratic equation, avoiding the need to factor (which can be difficult for complex or non-factorable quadratics). Using this formula ensures accuracy and deepens algebra comprehension.", "## Conclusion", "Solving ( t^2 - 15t + 56 = 0 ) by the quadratic formula yields two real solutions:", "[\n\boxed{t = 7} \quad \ ext{and} \quad \boxed{t = 8}\n]", "Mastering this method equips you with a powerful tool for tackling quadratic equations efficiently and confidently. Whether in school assignments, job interviews, or real-world applications, the quadratic formula remains an essential skill in your mathematical toolkit.", "---", "Keywords: quadratic formula, solve quadratic equation, t² - 15t + 56 = 0, quadratic roots, algebra tutorial, discriminant, real solutions, quadratic equations explanation.\nMeta description: Learn how to solve ( t^2 - 15t + 56 = 0 ) using the quadratic formula with step-by-step instructions, including discriminant calculation and verification."]