Solve the quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 25, c = -84 \).

["# Solving Quadratic Equations: A Step-by-Step Guide Using the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, essential for students and math enthusiasts alike. Whether you're tackling problems in the classroom, homework, or real-world applications, mastering the quadratic formula provides a reliable and efficient method to find solutions. In this article, we’ll explore how to solve the quadratic equation ( x^2 + 25x - 84 = 0 ) using the quadratic formula, especially with values ( a = 1 ), ( b = 25 ), and ( c = -84 ).", "## What is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation of the form:", "[\nax^2 + bx + c = 0\n]", "where ( a <br/>\neq 0 ). The roots (or solutions) of the equation can be real or complex, depending on the discriminant ( D = b^2 - 4ac ).", "## The Quadratic Formula: The Key to Solutions", "The quadratic formula provides a direct way to find the solutions ( x ) for any quadratic equation:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula works universally, making it ideal for equations where factoring is difficult or impossible.", "---", "## Step-by-Step Solution for ( x^2 + 25x - 84 = 0 )", "Let’s apply the quadratic formula to our specific equation. First, identify the coefficients:", "- ( a = 1 )\n- ( b = 25 )\n- ( c = -84 )", "### Step 1: Calculate the discriminant ( D )", "[\nD = b^2 - 4ac = (25)^2 - 4(1)(-84) = 625 + 336 = 961\n]", "Since ( D = 961 > 0 ), there are two distinct real solutions.", "### Step 2: Plug values into the quadratic formula", "[\nx = \frac{-25 \pm \sqrt{961}}{2(1)} = \frac{-25 \pm 31}{2}\n]", "### Step 3: Solve for both solutions", "Using the ( \pm ) sign:", "[\nx_1 = \frac{-25 + 31}{2} = \frac{6}{2} = 3\n]", "[\nx_2 = \frac{-25 - 31}{2} = \frac{-56}{2} = -28\n]", "---", "## Final Answer", "The solutions to the equation ( x^2 + 25x - 84 = 0 ) are:", "[\nx = 3 \quad \ ext{and} \quad x = -28\n]", "---", "## Why Use the Quadratic Formula?", "- Universal applicability: Works for any quadratic equation, even when ( a <br/>\neq 1 ) or when the equation doesn’t factor neatly.\n- Guaranteed solutions: Provides real or complex roots clearly and systematically.\n- Educational value: Reinforces algebraic understanding and precision in computation.", "---", "## Summary", "Solving a quadratic equation like ( x^2 + 25x - 84 = 0 ) becomes effortless with the quadratic formula. With ( a = 1 ), ( b = 25 ), and ( c = -84 ), we found real solutions ( x = 3 ) and ( x = -28 ) by carefully computing the discriminant and applying the formula. Whether you're preparing for exams or solving practical problems, mastering this method empowers you to tackle quadratics confidently.", "If you're ready to solve more equations, practice calculating the discriminant and applying the quadratic formula confidently — you're well on your way to mastering algebra!", "---", "Keywords: quadratic equation, quadratic formula, solve quadratic, solve ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), solve ( x^2 + 25x - 84 = 0 ), step-by-step solution, algebraic methods."]