\( x = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4} \).

["# Solving the Quadratic Equation: A Step-by-Step Guide to ( x = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4} )", "Solving quadratic equations is a fundamental skill in algebra, arising in countless applications across science, engineering, and economics. One classic form is the standard quadratic equation:\n[\nax^2 + bx + c = 0\n]\nToday, we dive into a specific method for solving such equations by rewriting them into a solvable format—specifically, the quadratic formula derived from completing the square. We’ll solve the equation:\n[\nx = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\n]\n...and unpack the steps clearly and effectively.", "---", "## Understanding the Components of the Quadratic Formula", "The quadratic formula is:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nThis formula gives the exact solutions for any quadratic equation. Let’s verify how it’s derived and apply it step-by-step to the expression above.", "---", "## Step 1: Identify ( a ), ( b ), and ( c ) from Standard Form", "Given the expression:\n[\nx = \frac{-(-4) \pm \sqrt{64}}{4}\n]\nwe can extract the coefficients by analyzing the numerator.", "- The term outside the square root, (-(-4)), implies ( b = -(-4) = 4 )\n- The number under the square root, ( 64 ), is ( b^2 - 4ac ), so ( b^2 - 4ac = 64 )\n- The denominator is ( 4 ), so ( 2a = 4 \Rightarrow a = 2 )", "Now, compute ( c ) to confirm:", "From ( b^2 - 4ac = 64 ):\n[\n4^2 - 4(2)c = 64 \Rightarrow 16 - 8c = 64 \Rightarrow -8c = 48 \Rightarrow c = -6\n]\nWait — here we see a consistency check: earlier, ((-4)^2 = 16), and from the formula (b^2 - 4ac = 64), so:\n[\n16 - 4(2)c = 64 \Rightarrow -8c = 48 \Rightarrow c = -6\n]\nBut in the simplified expression, only ( b ) and the radical are explicitly given, so let’s re-express the full equation to clarify.", "---", "## Step 2: From Standard Form to the Simplified Expression", "The original quadratic equation can be rewritten as:\n[\nx^2 + 4x + c = 0\n]\nsince ( -b = -(-4) = 4 ).", "To complete the square, rewrite:\n[\nx^2 + 4x = -c\n]\nAdd ( \left(\frac{4}{2}\right)^2 = 4 ) to both sides:\n[\nx^2 + 4x + 4 = 4 - c \Rightarrow (x + 2)^2 = 4 - c\n]\nTaking square roots:\n[\nx + 2 = \pm \sqrt{4 - c} \Rightarrow x = -2 \pm \sqrt{4 - c}\n]", "But here, we are told the expression results in:\n[\nx = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\n]\nThis implies ( b^2 - 4ac = 64 ), and if ( 2a = 4 ), then ( a = 2 ). Therefore, the actual equation must have been:\n[\n2x^2 + 4x + c = 0\n]\nwith ( b = 4 ), ( a = 2 ), and ( b^2 - 4ac = 64 \Rightarrow 16 - 8c = 64 \Rightarrow c = -6 ), so:\n[\n2x^2 + 4x - 6 = 0\n]\nDivide entire equation by 2:\n[\nx^2 + 2x - 3 = 0\n]\nWait—this gives ( b = 2 ), not 4. So how do we reconcile?", "Actually, the original expression ( x = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4} ) suggests the numerator corresponds to (-b \pm \sqrt{b^2 - 4ac}). So:", "- (-b = 4 \Rightarrow b = -4)\n- ( \sqrt{b^2 - 4ac} = \sqrt{64} = 8 \Rightarrow b^2 - 4ac = 64 )\n- Denominator is ( 2a = 4 \Rightarrow a = 2 )", "But then ( b = -4 ), and (-b = 4), matching the numerator. So the quadratic is:\n[\n2x^2 - 4x + c = 0\n]\nFrom ( b^2 - 4ac = 64 \Rightarrow (-4)^2 - 4(2)c = 64 \Rightarrow 16 - 8c = 64 \Rightarrow -8c = 48 \Rightarrow c = -6 )\nThus, the full equation is:\n[\n2x^2 - 4x - 6 = 0\n]\nDivide by 2:\n[\nx^2 - 2x - 3 = 0\n]", "Now the equation is clearly:\n[\nx^2 - 2x - 3 = 0,\quad a = 1, b = -2, c = -3\n]\nBut earlier steps imply ( b = -4 )—this suggests a mismatch unless we reinterpret the expression carefully.", "Actually, the form\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nis applied to the solved root of an equation already simplified. From the derivation above, if the correct simplified form leads to:\n[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-3)}}{2(2)} = \frac{4 \pm \sqrt{16 + 24}}{4} = \frac{4 \pm \sqrt{40}}{4}\n]\nWait — ( \sqrt{64} ) implies ( b^2 - 4ac = 64 ), so if ( b = -4 ), then ( (-4)^2 = 16 <br/>\ne 64 ). There’s inconsistency.", "Resolution: The expression ( x = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4} ) must be the solved result, not necessarily corresponding directly to ( a, b, c ) of original form. Rather, it's the final expression after substitution and solving.", "Let’s assume the original quadratic equation, after proper manipulation, simplifies to have:\n[\na = 2, \quad b = -4, \quad \ ext{and} \quad \sqrt{b^2 - 4ac} = \sqrt{64}\n]\nThen:\n[\n16 - 8c = 64 \Rightarrow -8c = 48 \Rightarrow c = -6\n]\nThus, the equation is:\n[\n2x^2 - 4x - 6 = 0\n]\nNow apply the quadratic formula:", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)} = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\n]\nThis matches the given expression.", "---", "## Step 3: Solving Using the Derived Formula", "Start with:\n[\nx = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\n]\nThis splits into two solutions:", "1. ( x = \frac{4 + 8}{4} = \frac{12}{4} = 3 )\n2. ( x = \frac{4 - 8}{4} = \frac{-4}{4} = -1 )", "So the solutions are ( x = 3 ) and ( x = -1 ).", "---", "## Why This Method Works", "The quadratic formula simplifies the lengthy completing-the-square procedure into a quick formula. When:\n- ( a = 2 )\n- ( b = -4 ) (so (-b = 4))\n- ( \sqrt{b^2 - 4ac} = \sqrt{64} \Rightarrow 16 - 4(2)c = 64 \Rightarrow c = -6 )", "This transformation allows immediate application of the solved form, even without fully completing the square.", "---", "## Practical Applications", "Quadratic equations model real-world phenomena like projectile motion, profit maximization, and circuit behavior. Mastering both derivation and shortcut formula usage strengthens problem-solving flexibility. Whether you’re testing a robot’s trajectory or analyzing business growth, knowing how to extract ( a ), ( b ), ( c ) and apply:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nis essential.", "---", "## Summary", "The equation\n[\nx = \frac{-(-4) \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\n]\nrepresents the two solutions to a quadratic equation derived via coefficient analysis and simplification. By recognizing ( b = -4 ), ( \sqrt{b^2 - 4ac} = \sqrt{64} ), and ( 2a = 4 ), we reconstruct the full quadratic and solve efficiently. Mastery of this process unlocks deeper algebraic insight and practical problem-solving power.", "---", "### Key Takeaways:\n- Verify coefficients (( a, b, c )) from the simplified expression.\n- Confirm values satisfy ( b^2 - 4ac = 64 ).\n- Apply the quadratic formula directly for rapid solutions.\n- This method bridges hand calculation and formula use.", "Perfect for students, educators, and anyone refining algebraic fluency!", "---", "### Related Search Terms:\n- How to solve quadratic equations by formula\n- Derive quadratic formula step by step\n- Solve ( x = \frac{4 \pm 8}{4} ) meaning\n- Quadratic equation solution practice problems\n- Understanding ( b^2 - 4ac = 64 ) in quadratics", "---", "Boost your algebra skills—solve smart, solve fast!"]