\( x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4} \).
["# Understanding Quadratic Equations: Solving ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) with ( x = \frac{-(-4) \pm \sqrt{64}}{2 \ imes 2} )", "Solving quadratic equations is a fundamental skill in algebra that appears across science, engineering, economics, and many real-world applications. One of the most powerful formulas for finding the solutions of any quadratic equation ( ax^2 + bx + c = 0 ) is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In this article, we will break down how to apply this formula to solve a specific quadratic equation:\n[\nx = \frac{-(-4) \pm \sqrt{64}}{2 \ imes 2} = \frac{4 \pm 8}{4}\n]", "## The Context Behind the Formula", "The quadratic formula stems from completing the square and provides the exact solutions to any equation of the form ( ax^2 + bx + c = 0 ). The discriminant—( \Delta = b^2 - 4ac )—tells us about the nature of the roots: whether they are real, repeated, or complex.", "## Step-by-Step Breakdown of the Given Equation", "Starting from the general form and substituting relevant values:", "### Step 1: Identify coefficients\nGiven the simplified expression:\n[\nx = \frac{-(-4) \pm \sqrt{64}}{2 \ imes 2}\n]", "Here, compar会議:\n- ( b = -4 )\n- Discriminant ( \sqrt{64} = 8 )\n- Denominator ( 2a ), and since ( 2 \ imes 2 = 4 ), we have ( a = 2 )", "### Step 2: Plug into the quadratic formula\n[\nx = \frac{ -(-4) \pm \sqrt{64} }{ 2 \ imes 2 } = \frac{4 \pm 8}{4}\n]", "### Step 3: Solve both cases\nThe ( \pm ) means we compute two possible values:", "- First solution (plus sign):\n ( x = \frac{4 + 8}{4} = \frac{12}{4} = 3 )", "- Second solution (minus sign):\n ( x = \frac{4 - 8}{4} = \frac{-4}{4} = -1 )", "### Final Solutions\nThe equation ( x^2 - 4x + 0 = 0 ) (since ( b = -4 ), ( c = 0 ), so ( x^2 - 4x = 0 )) has two real roots:", "[\nx = 3 \quad \ ext{and} \quad x = -1\n]", "## Key Takeaways", "- Always identify ( a ), ( b ), and ( c ) carefully from the quadratic equation.\n- The ( \pm ) symbol reflects that quadratic equations can have two distinct solutions.\n- When the discriminant ( b^2 - 4ac ) is positive, like here (( 64 )), we get two real roots.\n- Simplifying expressions before applying the formula saves errors and speed up solving.", "## Why Learning This Matters", "Understanding how to solve quadratic equations unlocks the ability to model a wide range of phenomena—from projectile motion and company profit maximization to physics and engineering optimizations. Mastering the quadratic formula equips you with a go-to tool for solving complex real-world problems.", "---", "Key SEO Keywords: quadratic formula, solving quadratic equations, discriminant, algebraic solutions, real vs complex roots, step-by-step equation solving, ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), step-by-step quadratic calculation, solving ( x = \frac{-(-4) \pm \sqrt{64}}{2 \ imes 2} )", "---", "Next Steps: Practice applying the quadratic formula with different coefficients. Try calculating the roots manually and verify using graphing technology or number substitution to confirm correctness. With regular practice, quadratic solutions will become intuitive and fundamental in your algebra toolkit!"]
