\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 1 \times (-21)}}{2 \times

["# Solving Quadratic Equations: A Step-by-Step Guide to the Quadratic Formula", "Understanding how to solve quadratic equations is a fundamental skill in algebra. Whether you're determining the roots of a polynomial or analyzing parabolic motion, mastering the quadratic formula unlocks a wide range of mathematical applications. In this guide, we’ll break down the quadratic equation ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), walk through its components, and show you how to use it correctly—even when values are negative or obscure.", "## What Is the Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation in standard form:", "[\nax^2 + bx + c = 0, \quad \ ext{where } a <br/>\neq 0\n]", "The general solution to this equation involves the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula provides the roots (or solutions) of the equation, which may be real or complex depending on the discriminant.", "---", "## Step-by-Step Breakdown of the Formula", "To apply the formula correctly, let’s examine each part:", "- ( a, b, c ): Coefficients from the quadratic equation ( ax^2 + bx + c = 0 )\n- ( -b ): Negative coefficient of ( x ), used because solving – not expanding – the equation\n- ( b^2 - 4ac ): The discriminant—critical for determining the nature of the roots\n- ( \pm ): Indicates two possible solutions—one using +, one using –\n- ( \sqrt{\ ext{expression}} ): The square root symbol; requires the expression under it (discriminant) to be non-negative for real roots\n- ( 2a ): Denominator ensures the equation is properly scaled to isolate ( x )", "---", "### Example Problem: ( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \ imes 1 \ imes (-21)}}{2 \ imes 1} )", "Let’s use this template with concrete numbers from a typical quadratic expression.", "#### Step 1: Identify coefficients\nGiven:\n- ( a = 1 )\n- ( b = -4 )\n- ( c = -21 )", "#### Step 2: Plug values into the formula", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-21)}}{2(1)}\n]", "#### Step 3: Simplify each term\n- ( -(-4) = +4 )\n- ( b^2 = (-4)^2 = 16 )\n- ( 4ac = 4 \ imes 1 \ imes (-21) = -84 ), so ( -4ac = +84 )\n- Discriminant: ( 16 + 84 = 100 )\n- Denominator: ( 2 \ imes 1 = 2 )", "Thus:", "[\nx = \frac{4 \pm \sqrt{100}}{2}\n]", "[\nx = \frac{4 \pm 10}{2}\n]", "#### Step 4: Compute both roots", "- First root (( + )):\n[\nx = \frac{4 + 10}{2} = \frac{14}{2} = 7\n]", "- Second root (( - )):\n[\nx = \frac{4 - 10}{2} = \frac{-6}{2} = -3\n]", "✅ Final solutions: ( x = 7 ) or ( x = -3 )", "---", "## Understanding the Discriminant", "The discriminant ( D = b^2 - 4ac ) determines the type of solutions:", "| Discriminant ( D ) | Meaning |\n|----------------------|-----------------------------------|\n| ( D > 0 ) | Two distinct real roots |\n| ( D = 0 ) | One real double root |\n| ( D < 0 ) | Two complex conjugate roots |", "In our example, ( D = 100 > 0 ), confirming two real solutions.", "---", "## Why the Format Matters", "Notice how the negative sign before ( b ) and the negative denominator ( 2a ) are crucial. Forgetting either changes the result:", "- Missing the negative on ( b ): ( x = \frac{-b \pm \cdots} ) becomes ( x = \frac{+b \pm \cdots} )\n- Incorrect denominator: using ( a ) instead of ( 2a ) leads to errors in scaling.", "Always double-check signs and coefficients.", "---", "## More Than Just Answers: Applications of the Quadratic Formula", "Solving quadratics isn’t just about finding roots—it’s about:", "- Analyzing parabolas in physics (e.g., projectile motion)\n- Designing curves in engineering and architecture\n- Optimizing profits in economics\n- Solving geometric problems involving areas and distances", "Understanding the quadratic formula prepares you for these real-world applications.", "---", "## Practice Problem", "Test your knowledge with this equation:", "[\nx = \frac{-3 \pm \sqrt{9 + 36}}{6}\n]", "- Identify ( a = 1 ), ( b = 3 ), ( c = 36 )\n- Calculate the discriminant\n- Simplify and write both roots", "✅ Hint: ( b^2 = 9 ), ( 4ac = 4 \ imes 1 \ imes 36 = 144 ), so discriminant = ( 9 - 144 = -135 )\n→ Complex roots!", "---", "## Conclusion", "Mastering the quadratic formula equips you to solve any second-degree equation confidently. By following the structured approach—identifying coefficients, applying signs carefully, and computing the discriminant—you ensure accuracy and deepen conceptual understanding. Whether you're a student, educator, or self-learner, regular practice with varied examples strengthens your algebraic fluency.", "Start applying this method today—your next equation is just a formula away!", "---", "Keywords: quadratic equation, quadratic formula, solving quadratics, discriminant, algebra, real roots, complex roots, quadratic formula example, math tutorial, high school algebra, quadratic formula steps\nMeta Description: Learn how to solve ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) with step-by-step explanation, including sign handling, discriminant significance, and practical applications. Ideal for students and educators."]