Use quadratic formula: x = [−2 ± √(4 + 44)] / 2 = [−2 ± √48]/2 = [−2 ± 4√3]/2 = −1 ± 2√3.

["Understanding the Quadratic Formula: A Step-by-Step Guide with x = −1 ± 2√3", "Solving quadratic equations is a fundamental skill in algebra, and the quadratic formula is an essential tool for finding precise solutions quickly. In this article, we’ll walk through the classic quadratic equation using the formula and specifically explore the elegant solution expressed as x = −1 ± 2√3. Whether you're a student, teacher, or self-learner, mastering this process will clarify how to tackle quadratic expressions with confidence.", "---", "### What is the Quadratic Formula?", "The quadratic formula helps solve equations in the standard form:", "[\nax^2 + bx + c = 0\n]", "The formula is:\n[\nx = \frac{−b ± \sqrt{b^2 − 4ac}}{2a}\n]", "This formula gives the two possible solutions (roots) of the quadratic equation, depending on the ± sign. When the discriminant ( b^2 − 4ac ) is positive, there are two real roots; when zero, one real root; and when negative, complex roots.", "---", "### Applying the Formula: Step-by-Step Example", "Consider a typical quadratic equation where ( a = 1 ), ( b = 2 ), and ( c = 11 ), so the equation is:", "[\nx^2 + 2x + 11 = 0\n]", "Now, substitute ( a ), ( b ), and ( c ) into the quadratic formula:", "[\nx = \frac{−(2) ± √(2^2 − 4(1)(11))}{2(1)}\n]", "Simplify step by step:", "- Calculate the discriminant:\n [\n b^2 − 4ac = 4 − 44 = −40\n ]", "- Since the discriminant is negative (−40), we know the roots are complex. But let’s check how simplification applies algebraically.", "Still, complete the formula symbolically for clarity:", "[\nx = \frac{−2 ± √(4 − 44)}{2} = \frac{−2 ± √(−40)}{2}\n]", "Factor the square root:", "[\n√(−40) = √(−1 \ imes 4 \ imes 10) = √(4) \cdot √(−10) = 2i\sqrt{10}\n]", "Note: Though in this example the roots are complex, the algebraic manipulation follows the standard form and leads to simplification features seen in simpler real-root cases.", "However, back to your original focus — simplifying to:", "[\nx = −1 ± 2\sqrt{3}\n]", "This elegant form arises when the discriminant fits a perfect square under the radical. Let’s verify how such simplified radical forms emerge through careful substitution and factoring.", "---", "### Why Domain Simplifies to x = −1 ± 2√3?", "If we re-analyze the quadratic in a way that mimics your expression — that is, arriving logically at ( x = −1 ± 2\sqrt{3} )—we might start from an equation like:", "[\nx^2 + 2x + 13 = 0\n]", "Now apply the quadratic formula:", "- ( a = 1 ), ( b = 2 ), ( c = 13 )", "[\nx = \frac{−2 ± √(2^2 − 4(1)(13))}{2} = \frac{−2 ± √(4 − 52)}{2} = \frac{−2 ± √(−48)}{2}\n]", "Even in this case, the discriminant is negative. However, suppose the intended context involves solving a modified or scaled quadratic for pattern recognition, such as in physics or geometry, where solution forms simplify beautifully.", "For the exact expression ( x = −1 ± 2\sqrt{3} ), consider that:", "[\n−1 ± 2\sqrt{3} = −1 ± 2 \ imes 1.732 \approx −1 ± 3.464\n]", "These approximate values (~−4.464 and ~2.464) suggest roots arising from a quadratic with roots differing by 4√3 — common in parabolic motion or design problems where symmetry or vertex form offers insight.", "---", "### Practical Applications of Quadratic Solutions", "The formula’s utility spans multiple domains:", "- Physics: Modeling projectile motion, where time of flight depends on solving quadratic equations for maximum range.\n- Engineering: Calculating stress points and structural integrity referenced through quadratic relationships.\n- Economics: Profit and revenue optimization often involve quadratic cost or revenue models.", "Understanding exact solutions like x = −1 ± 2√3 enables deeper insight into these applications, especially when symbolic manipulation reveals root behavior or symmetry.", "---", "### Final Thoughts", "Mastering the quadratic formula goes beyond rote calculation — it builds algebraic intuition and problem-solving agility. While the form ( x = −1 ± 2\sqrt{3} ) may shine in educational contexts or pattern-based learning to illustrate roots derived from simplified discriminants, real-world applications often require combining formulas with factoring, completing the square, or using graphing tools.", "By internalizing each step — from identifying ( a ), ( b ), ( c ) to simplifying radicals — learners empower themselves to confidently tackle equations and connect mathematics to meaningful real-world phenomena.", "---", "Remember:\nIf you encounter a quadratic equation that simplifies to ( x = −1 ± 2\sqrt{3} ), assess the coefficients carefully—this typical pattern suggests a discriminant ( b^2 − 4ac = 12 ), leading to ( ±\sqrt{12} = ±2\sqrt{3} ) in simplified radical form.", "Keep practicing, stay curious, and let the quadratic formula be your trusted companion in algebraic mastery!"]