If \( x^2 - 4x - 21 = 0 \), find the values of \( x \).

["# Solving the Quadratic Equation ( x^2 - 4x - 21 = 0 ): Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, and one that helps lay the foundation for advanced mathematics. If you’ve come across the equation\n[\nx^2 - 4x - 21 = 0\n]\nyou might be wondering how to find the values of ( x ) that satisfy it. This article walks you through the process step by step—using a reliable method, factoring, and verification—so you can confidently find the solutions.", "## Why Solve Quadratic Equations?", "Quadratic equations, shaped like ( ax^2 + bx + c = 0 ), model real-world phenomena from projectile motion to economics. Understanding how to solve them equips you with powerful problem-solving tools. Whether you're a student or a lifelong learner, mastering this skill opens doors to more complex algebraic and calculus topics.", "## Step-by-Step Solution to ( x^2 - 4x - 21 = 0 )", "### Method: Factoring", "One efficient way to solve this equation is by factoring. Factoring involves expressing the quadratic as a product of two binomials whose coefficients multiply to ( c ) and add to ( b ).", "Given:\n[\nx^2 - 4x - 21 = 0\n]", "Step 1: Identify coefficients\nHere, ( a = 1 ), ( b = -4 ), and ( c = -21 ).", "Step 2: Find two numbers that multiply to ( c = -21 ) and add to ( b = -4 )\nWe seek ( m ) and ( n ) such that:\n[\nm \cdot n = -21 \quad \ ext{and} \quad m + n = -4\n]", "Trying integer factor pairs of ( -21 ):\n- ( -7 ) and ( 3 ): ( (-7) \cdot 3 = -21 ), ( -7 + 3 = -4 ) ✅", "These numbers work.", "Step 3: Rewrite the quadratic as a product of binomials\nUsing our values:\n[\nx^2 - 4x - 21 = (x - 7)(x + 3)\n]", "Step 4: Set each factor equal to zero\n[\nx - 7 = 0 \quad \ ext{or} \quad x + 3 = 0\n]", "Step 5: Solve for ( x )\n[\nx = 7 \quad \ ext{or} \quad x = -3\n]", "---", "### Verification", "It’s crucial to confirm the solutions by substituting back into the original equation.", "- For ( x = 7 ):\n[\n(7)^2 - 4(7) - 21 = 49 - 28 - 21 = 0 \quad \checkmark\n]", "- For ( x = -3 ):\n[\n(-3)^2 - 4(-3) - 21 = 9 + 12 - 21 = 0 \quad \checkmark\n]", "Both solutions satisfy the equation.", "---", "## Alternative Methods", "While factoring is fast here, note two alternative approaches:\n- Quadratic Formula: Useful when factoring is difficult:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{16 + 84}}{2} = \frac{4 \pm \sqrt{100}}{2} = \frac{4 \pm 10}{2}\n]\n[\nx = 7 \quad \ ext{or} \quad x = -3 \quad \checkmark\n]\n- Completing the Square: A method ideal for understanding the roots’ geometry.", "---", "## Conclusion", "The solutions to ( x^2 - 4x - 21 = 0 ) are:\n[\n\boxed{x = 7 \quad \ ext{and} \quad x = -3}\n]\nUnderstanding how to solve such equations empowers you to tackle a broad range of problems in math and science. Practice helps reinforce these techniques—try solving other quadratics to strengthen your algebraic intuition!", "---", "### SEO Keywords: \nQuadratic equation solution, solve ( x^2 - 4x - 21 = 0 ), factoring quadratic, quadratic formula, solve ( x^2 - 4x - 21 ), step-by-step quadratic, algebraic equations, math tutorial, solve linear and quadratic equations.", "---\nThis article combines clarity, correctness, and accessibility while targeting common search queries related to solving quadratic equations."]