Solve for \(x\) in the quadratic equation \(2x^2 - 3x - 5 = 0\).
["# Solve for ( x ) in the Quadratic Equation: ( 2x^2 - 3x - 5 = 0 )", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and STEM professionals alike. One commonly encountered quadratic equation is ( 2x^2 - 3x - 5 = 0 ). This article guides you step-by-step through solving for ( x ) using the quadratic formula, offering clear explanations and practical insight.", "## Understanding the Quadratic Equation", "A quadratic equation always takes the standard form:", "[\nax^2 + bx + c = 0\n]", "For the equation ( 2x^2 - 3x - 5 = 0 ), identifying coefficients gives:", "- ( a = 2 )\n- ( b = -3 )\n- ( c = -5 )", "The values of ( a ), ( b ), and ( c ) are crucial as they determine the method of solving and the nature of the roots.", "## Using the Quadratic Formula", "Since factoring isn’t immediately obvious, the most reliable method is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "### Step 1: Calculate the Discriminant", "The discriminant ( D = b^2 - 4ac ) determines the number and type of solutions:", "[\nD = (-3)^2 - 4(2)(-5) = 9 + 40 = 49\n]", "Since ( D = 49 > 0 ), there are two distinct real solutions.", "### Step 2: Substitute Coefficients into the Formula", "Now plug ( a = 2 ), ( b = -3 ), and ( D = 49 ) into the quadratic formula:", "[\nx = \frac{-(-3) \pm \sqrt{49}}{2(2)} = \frac{3 \pm 7}{4}\n]", "### Step 3: Solve for Both Roots", "Compute the two possible values of ( x ):", "1. ( x_1 = \frac{3 + 7}{4} = \frac{10}{4} = \frac{5}{2} )\n2. ( x_2 = \frac{3 - 7}{4} = \frac{-4}{4} = -1 )", "## Final Answer", "The solutions to the equation ( 2x^2 - 3x - 5 = 0 ) are:", "[\n\boxed{x = \frac{5}{2} \quad \ ext{and} \quad x = -1}\n]", "## Why This Equation Matters", "Understanding how to solve quadratic equations is vital in physics, engineering, economics, and computer science. From projectile motion calculations to optimizing profit models, quadratic equations are foundational tools in modeling real-world situations.", "By mastering techniques like the quadratic formula and discriminant analysis, learners transform abstract math into practical problem-solving skills.", "## Summary", "- Given: ( 2x^2 - 3x - 5 = 0 )\n- Identified: ( a = 2 ), ( b = -3 ), ( c = -5 )\n- Discriminant: ( D = 49 > 0 ) → two real solutions\n- Solutions: ( x = \frac{5}{2} ) and ( x = -1 )", "Whether you're a student preparing for exams or a professional brushing up on foundational math, knowing how to solve ( 2x^2 - 3x - 5 = 0 ) empowers you with a key algebraic tool.", "---", "Keywords: solve for (x) in (2x^2 - 3x - 5 = 0), quadratic equation solutions, quadratic formula, discriminant, algebra practice, real and distinct roots, step-by-step quadratic formula, how to solve quadratic equations."]
