Use quadratic formula: \( x = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2} = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3} \)

["# Solve Quadratic Equations Like a Pro: Mastering the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and anyone working with mathematical models. Whether you’re tackling a textbook problem or analyzing real-world data, using the quadratic formula ensures accurate and efficient results. In this article, we walk through a classic quadratic equation step-by-step, demonstrating how the quadratic formula transforms a challenging quadratic expression into two clear solutions.", "---", "## What Is the Quadratic Formula?", "The quadratic formula is a powerful formula used to solve equations of the form:\n[ ax^2 + bx + c = 0 ]\nThe formula is:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula provides the two (possibly equal) solutions for ( x ) — the roots of the equation — regardless of whether the coefficients ( a ), ( b ), and ( c ) are integers, fractions, or irrational numbers.", "---", "## Example Problem", "Consider the quadratic equation:\n[\nx^2 + 2x + 13 = 0\n]", "At first glance, the coefficients are ( a = 1 ), ( b = 2 ), and ( c = 13 ). Let’s solve it using the quadratic formula.", "---", "## Step-by-Step Solution", "Start by identifying the coefficients:\n[\na = 1,\quad b = 2,\quad c = 13\n]", "Plug these into the standard formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\n[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}\n]", "Calculate the discriminant ( b^2 - 4ac ):\n[\n400 + 4 \cdot 1 \cdot 13 = 4 + 52 = 56 \Rightarrow \sqrt{4 + 52} = \sqrt{56}\n]", "But note: the original problem simplifies the square root expression beautifully. Observe:\n[\nb^2 - 4ac = 4 + 52 = 56 \Rightarrow \sqrt{56} = \sqrt{4 \cdot 13} = 2\sqrt{13}\n]\nWait — correction: earlier assertion of ( \sqrt{48} ) or ( \sqrt{4 + 44} ) was incorrect. Let’s re-evaluate:", "Actually, note the original problem states:\n[\nx = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2} = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "Here, ( b = 2 \Rightarrow b^2 = 4 ), and ( 4ac = 4 \cdot 1 \cdot 13 = 52 ), but to get ( 48 ), something must be off unless ( c = 12 ), not 13.", "But wait — let's resolve this carefully.", "If instead, the equation is:\n[\nx^2 + 2x + 12 = 0\n]", "Then ( a = 1 ), ( b = 2 ), ( c = 12 ), so:\n[\nb^2 - 4ac = 4 - 48 = -44 \quad \ ext{(not real)}\n]", "So only equations yielding ( \sqrt{48} ) or ( 4\sqrt{3} ) involve constants such as ( x^2 + 2x + 13 = 0 ), but that discriminant is ( 4 - 52 = -48 ), resulting in imaginary roots.", "Instead, try a corrected example matching the proposed solution.", "Let’s solve:\n[\nx^2 + 2x + 13 = 0\n]", "Discriminant:\n[\nb^2 - 4ac = 2^2 - 4(1)(13) = 4 - 52 = -48\n]\n→ Complex roots. Not ideal for simplicity.", "But suppose the equation is:\n[\nx^2 + 4x + 13 = 0\n]", "Then ( a = 1 ), ( b = 4 ), ( c = 13 ) →\n[\nb^2 - 4ac = 16 - 52 = -36\n]", "Still complex.", "Wait — notice the expression in the original problem:\n[\nx = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2}\n]", "So ( b^2 - 4ac = 48 ). Thus, ( b^2 = 4 + 44 = 48 \Rightarrow b = \sqrt{48} = 4\sqrt{3} ). This implies ( b = 4\sqrt{3} ), which is unusual unless ( b ) is irrational.", "But in standard quadratic equations, ( b ) is the linear coefficient, usually a rational or integer number. So likely, the equation is set up so that ( b^2 - 4ac = 48 ), with ( b = \pm 2 ), which contradicts — unless misinterpreted.", "Wait — there is a mismatch between the formula and the arithmetic.", "Let’s reconstruct a proper example.", "---", "### Correct Simplified Example", "Let’s use the equation:\n[\nx^2 + 4x + 13 = 0\n]", "Identify:\n- ( a = 1 )\n- ( b = 4 )\n- ( c = 13 )", "Compute discriminant:\n[\nb^2 - 4ac = 4^2 - 4(1)(13) = 16 - 52 = -36\n]", "Still complex.", "But suppose:\n[\nx^2 + 2x + 1 = 0\n]\n→ perfect square: ( (x+1)^2 = 0 ), roots ( x = -1 )", "Too simple.", "To align with the given simplified radicals, consider:\nLet’s assume the equation is\n[\nx^2 + 2x + 13 = 0\n]\nand accept discriminant is ( -48 ), giving complex roots:\n[\nx = \frac{-2 \pm \sqrt{-48}}{2} = \frac{-2 \pm 4\sqrt{3}i}{2} = -1 \pm 2\sqrt{3}i\n]\nBut the simplified result given shows real irrational ( -1 \pm 2\sqrt{3} ), meaning likely ( c = 13 ) was meant to be 12, or problem has typo.", "But for educational clarity, let’s correctly adapt the calc to match the expected answer.", "---", "## Fixing the Example for Clarity", "Let’s solve:\n[\nx^2 + 2x + 13 = 0\n]\n→ This yields complex roots, not matching ( -1 \pm 2\sqrt{3} ).", "Instead, solve:\n[\nx^2 + 4x + 7 = 0\n]", "Then:\n( a = 1 ), ( b = 4 ), ( c = 7 )\n( b^2 - 4ac = 16 - 28 = -12 ) → ( \sqrt{-12} ) — still complex.", "Try:\n[\nx^2 + 4x + 3 = 0\n]\n→ ( a = 1 ), ( b = 4 ), ( c = 3 )\n( b^2 - 4ac = 16 - 12 = 4 ) → roots: ( x = \frac{-4 \pm 2}{2} \Rightarrow -1, -3 )", "No irrationals.", "Only with ( b^2 - 4ac = 48 ) do we get ( \sqrt{48} = 4\sqrt{3} ).", "Let’s pick:\n[\nx^2 + 2x + 13 = 0 \Rightarrow b = 2 \Rightarrow b^2 = 4 <br/>\ne 48\n]", "So unless ( b = \sqrt{48} ), not matching.", "Conclusion: The expression\n[\nx = \frac{-2 \pm \sqrt{48}}{2} = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]\nis mathematically correct only if the quadratic coefficient ( b = 2 ), ( c = 13 ), and ( b^2 - 4ac = 48 \Rightarrow 4 - 4a \cdot 13 = 48 \Rightarrow -52a = 44 \Rightarrow a = -\frac{11}{13} ), a fractional leading coefficient.", "Such equations can occur in advanced contexts.", "---", "## Optimal Example to Match Final Answer", "Instead, solve:\n[\nx^2 - 2x + 13 = 0\n]\n→ ( a = 1 ), ( b = -2 ), ( c = 13 )", "Compute discriminant:\n[\nb^2 - 4ac = (-2)^2 - 4(1)(13) = 4 - 52 = -48\n] → complex", "Try:\n[\nx^2 + 4x + 13 = 0\n] → same", "Best choice:\nUse\n[\nx^2 + 4x + 12 = 0\n]\n→ ( b^2 - 4ac = 16 - 48 = -32 )", "Still negative.", "To get discriminant 48 and leading coefficient 1:\n[\nx^2 + bx + c = 0,\quad b^2 - 4c = 48\n]\nLet ( b = 2 ):\n[\n4 - 4c = 48 \Rightarrow -4c = 44 \Rightarrow c = -11\n] →\n[\nx^2 + 2x - 11 = 0\n]\nNow solve:\n[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11)}}{2} = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2} = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "Perfect match!", "---", "## Final Computation: The Full Solution", "Given:\n[\nx^2 + 2x - 11 = 0\n]", "With ( a = 1 ), ( b = 2 ), ( c = -11 ), discriminant:\n[\n\Delta = b^2 - 4ac = 4 - 4(1)(-11) = 4 + 44 = 48\n] → ( \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} )", "Then:\n[\nx = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-2 \pm 4\sqrt{3}}{2}\n]\nSimplify:\n[\nx = -1 \pm 2\sqrt{3}\n]", "---", "## Why This Formula Works", "The quadratic formula transforms a general ( ax^2 + bx + c = 0 ) into explicit solutions by balancing the square root of the discriminant, whether real or irrational. Recognizing patterns — like discriminants that yield ( \sqrt{48} ) — accelerates problem-solving in algebra.", "---", "## Real-World Applications", "Quadratic equations model real-life phenomena"]