Solve for \( x \) in the equation \( 2x^2 - 3x - 5 = 0 \) using the quadratic formula.
["# Solve for ( x ) in the Equation ( 2x^2 - 3x - 5 = 0 ) Using the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, essential for fields like physics, engineering, and data science. The equation ( 2x^2 - 3x - 5 = 0 ) is a classic example that demonstrates how to find exact solutions using the quadratic formula. In this article, we’ll guide you step-by-step through solving for ( x ), explain the quadratic formula, and highlight how this method applies to real-world problem solving.", "## Understanding the Quadratic Equation", "The standard form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "Where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). For the equation ( 2x^2 - 3x - 5 = 0 ), the coefficients are:", "- ( a = 2 )\n- ( b = -3 )\n- ( c = -5 )", "## The Quadratic Formula: A Reliable Solution Tool", "To solve for ( x ), we use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula guarantees real (and possibly complex) solutions regardless of whether the equation has two real roots, one real root, or two complex roots—depending on the discriminant ( \Delta = b^2 - 4ac ).", "### Step 1: Identify Coefficients", "First, plug in your values:", "- ( a = 2 )\n- ( b = -3 )\n- ( c = -5 )", "### Step 2: Calculate the Discriminant", "[\n\Delta = b^2 - 4ac = (-3)^2 - 4(2)(-5)\n]", "[\n\Delta = 9 + 40 = 49\n]", "A positive discriminant (( \Delta = 49 > 0 )) means two distinct real solutions exist.", "### Step 3: Apply the Quadratic Formula", "Now substitute ( a ), ( b ), and ( \Delta ) into the formula:", "[\nx = \frac{-(-3) \pm \sqrt{49}}{2(2)}\n]", "Simplify step by step:", "[\nx = \frac{3 \pm 7}{4}\n]", "### Step 4: Solve for Both Roots", "Compute each value:", "- First root (( + ) sign):", "[\nx = \frac{3 + 7}{4} = \frac{10}{4} = \boxed{2.5}\n]", "- Second root (( - ) sign):", "[\nx = \frac{3 - 7}{4} = \frac{-4}{4} = -1\n]", "## Final Answer", "The solutions to the equation ( 2x^2 - 3x - 5 = 0 ) are:", "[\nx = 2.5 \quad \ ext{and} \quad x = -1\n]", "## Why This Method Matters", "Using the quadratic formula ensures accuracy and symmetry in the process, making it easy to verify your work. Whether solving equations in school, programming math algorithms, or modeling physical systems, mastering this technique sharpens your analytical thinking and problem-solving efficiency.", "---", "Keywords: solve quadratic equation, quadratic formula, solve ( 2x^2 - 3x - 5 = 0 ), algebraic solutions, real roots, discriminant, math formula guide, step-by-step solving", "---", "Start mastering quadratic equations today—your next math challenge awaits!"]
